# Effective Electromagnetic Wave Properties of Disordered Stealthy
Hyperuniform Layered Media Beyond the Quasistatic Regime

Jaeuk Kim  Princeton Materials Institute, Princeton University, Princeton, New
Jersey 08544, USA  Department of Chemistry, Princeton University, Princeton,
New Jersey 08544, USA  Department of Physics, Princeton University, Princeton,
New Jersey 08544, USA  Salvatore Torquato  Princeton Materials Institute,
Princeton University, Princeton, New Jersey 08544, USA  Department of
Chemistry, Princeton University, Princeton, New Jersey 08544, USA  Department
of Physics, Princeton University, Princeton, New Jersey 08544, USA  Program in
Applied and Computational Mathematics, Princeton University, Princeton, New
Jersey 08544, USA<EMAIL_ADDRESS>
###### Abstract

_Disordered stealthy hyperuniform_ dielectric composites exhibit novel
electromagnetic wave transport properties in two and three dimensions. Here,
we carry out the first study of the electromagnetic properties of one-
dimensional (1D) disordered stealthy hyperuniform layered media. From an exact
nonlocal theory, we derive an approximation formula for the effective dynamic
dielectric constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q},\omega\right)}}$
of general 1D media that is valid well beyond the quasistatic regime and apply
it to 1D stealthy hyperuniform systems. We consider incident waves of
transverse polarization, frequency $\omega$, and wavenumber $k_{q}$. Our
formula for $\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{q},\omega\right)}}$, which is given in terms of the _spectral density_ ,
leads to a closed-form relation for the transmittance $T$. Our theoretical
predictions are in excellent agreement with finite-difference time-domain
(FDTD) simulations. Stealthy hyperuniform layered media have perfect
transparency intervals up to a finite wavenumber, implying no Anderson
localization, but non-stealthy hyperuniform media are not perfectly
transparent. Our predictive theory provides a new path for the inverse design
of the wave characteristics of disordered layered media, which are readily
fabricated, by engineering their spectral densities.

††journal: opticajournal

##  1 Introduction

_Disordered hyperuniform_ many-body systems [1, 2, 3] are an emerging class of
amorphous states of matter that are endowed with the novel wave and other
transport properties with advantages over their periodic counterparts [4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 3, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Such
hyperuniform two-phase composites are characterized by an anomalous
suppression of volume-fraction fluctuations in the infinite-wavelength limit
[2, 3], i.e., the local volume-fraction variance
$\mathinner{\sigma_{V}^{2}\mathopen{\left(R\right)}}$ inside a spherical
observation window of radius $R$ decays faster than $R^{-d}$ in $d$ dimensions
in the large-$R$ limit,
$\lim_{R\to\infty}R^{d}\mathinner{\sigma_{V}^{2}\mathopen{\left(R\right)}}=0.$
Equivalently, its associated _spectral density_
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}$ vanishes as
the wavenumber $\absolutevalue{{\bf k}}$ tends to zero, i.e.,
$\lim_{\absolutevalue{{\bf k}}\to
0}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}=0.$ One
important class of such hyperuniform media are the disordered _stealthy_
varieties in which $\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf
k}\right)}}=0$ for $0<\absolutevalue{{\bf k}}<K$ [24, 25, 26, 27], meaning
that they completely suppress single scattering of incident radiation for
these wavevectors [25, 3]. The degree of stealthiness $\chi$ is the ratio of
the number of the constrained wavevectors in the reciprocal space to the total
number of degrees of freedom. Recent studies showed that such exotic
disordered media exhibit novel electromagnetic wave transport properties,
including high transparency in the optically dense regime, maximized
absorption, and complete photonic band-gap formation [4, 5, 7, 8, 10, 12, 13,
15, 14, 28, 29, 16, 30, 17, 20, 21, 22, 23], in two and three spatial
dimensions. For example, previous work on light transparency of stealthy
hyperuniform systems considered 2D point scatterers [7] and 2D and 3D two-
phase dielectric composites [10, 31]. Here, we undertake the first study of
the electromagnetic properties of 1D disordered stealthy hyperuniform layered
media.

The problem of wave propagation in a two-phase (or multi-phase) layered medium
has been extensively studied, because of its simplicity and its ease of
fabrication [32, 33], including surface plasmons [34, 35], Anderson
localization [36, 37, 38, 39, 40, 41], and deep-subwavelength disorder [42],
as well as practical applications [43, 44, 18, 45]. An important wave
characteristic is the effective dynamic dielectric constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q},\omega\right)}}$
for the incident radiation of frequency $\omega$ and wavevector ${\bf k}_{q}$.
This complex-valued quantity determines the effective wavenumber $k_{e}$ and
extinction mean free path $\ell_{e}$. While there have been many
theoretical/numerical treatments to estimate
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q},\omega\right)}}$
of layered media [46, 47, 48, 49, 50, 51, 52, 53], previous approximations are
applicable to disordered media in the _quasistatic_ or long-wavelength regime,
i.e., $\absolutevalue{{\bf k}_{q}}\xi\ll 1$, where $\xi$ is a characteristic
inhomogeneity length scale.

Torquato and Kim [31] recently derived the general nonlocal strong-contrast
expansion for the effective dielectric constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q},\omega\right)}}$
that can be applied to two-phase media with various symmetries. This expansion
exactly treats multiple scattering to all orders beyond the quasistatic regime
(i.e., $0\leq\absolutevalue{{\bf k}_{q}}\xi\lesssim 1$) as a series involving
functionals of the $n$-point correlation functions
$\mathinner{S^{(i)}\mathopen{\left({\bf x}_{1},\ldots,{\bf x}_{n}\right)}}$
for all $n$ (Sec. 2). Here, the quantity
$\mathinner{S^{(i)}\mathopen{\left({\bf x}_{1},\ldots,{\bf x}_{n}\right)}}$
gives the probability of finding $n$ points at positions ${\bf
x}_{1},\ldots,{\bf x}_{n}$ all in phase $i\hskip 2.0pt(=1,2)$. Because of the
fast-convergence property of this series (or the linear fractional form of
this series), truncating it at the $n$-point level yields _multiple-
scattering_ approximations that still accurately capture multiple scattering
to all orders for a wide class of microstructures, including statistically
_anisotropic_ media. The second-order truncations already provide accurate
approximations for 2D and 3D statistically isotropic two-phase media [31].
However, and importantly, analogous approximations for statistically
anisotropic media have yet to be extracted and applied.

Here, we theoretically and numerically investigate the tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q},\omega\right)}}$
of 3D anisotropic layered media consisting of infinite parallel dielectric
slabs of phases 1 and 2 whose thicknesses are derived from 1D disordered
stealthy hyperuniform packings at various $\chi$ values. For simplicity, we
focus on normally incident waves of transverse polarization, where ${\bf
k}_{q}=k_{q}\hat{{\bf z}}$ (see Fig. 1), and thus the wavenumber $k_{q}$ is
the independent variable of the effective dielectric constant. From the exact
strong-contrast expansion, we derive, for the first time, formulas for
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{q},\omega\right)}}$ for 3D
anisotropic layered media that accurately accounts for multiple scattering in
terms of the spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}$, enabling
us to probe a wide range of wavenumbers. The quantity
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}$ is the
Fourier transform of the autocovariance function
$\mathinner{\chi_{{}_{V}}\mathopen{\left({\bf
r}\right)}}\equiv\mathinner{S_{2}^{(i)}\mathopen{\left({\bf
r}\right)}}-{\phi_{i}}^{2}$ [54], where ${\bf r}\equiv{\bf x}_{2}-{\bf
x}_{1}$, and can be measured from scattering experiments [55]. To our
knowledge, this expression is the first closed-form formula of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{q},\omega\right)}}$ for
general 1D media that applies well beyond the quasistatic regime.

We numerically verify that our derived formula can accurately capture multiple
scattering effects due to correlated disorder beyond the quasistatic regime by
using finite-difference time-domain (FDTD) simulations. For this purpose, we
numerically generate stealthy hyperuniform stratified dielectric two-phase
media via a modified collective-coordinate procedure described in Refs. [16,
31] (Sec. 3). For dimensionless wavenumbers up to $k_{1}/\rho\lesssim 1.5$,
our predictions indeed show excellent agreement with the real and imaginary
parts of the effective dielectric constant as well as transmittance found from
FDTD simulations (Secs. 4-5 of Supplement 1). Notably, our formula predicts
that stealthy hyperuniform layered media are perfectly transparent (defined as
$\imaginary[\mathinner{\varepsilon_{e}\mathopen{\left(k_{q},\omega\right)}}]=0$)
up to a finite wavenumber
$K_{T}=K/(2\sqrt{\phi_{1}\varepsilon_{1}+\phi_{2}\varepsilon_{2}})$ (i.e., no
Anderson localization) in the infinite-volume limit; see (12). This result is
especially remarkable because extended states in 1D disordered systems are
more difficult to achieve than in higher dimensions [38, 39, 40, 41, 17].

We also show that a perfect transparency interval cannot exist in disordered
1D _non-stealthy_ media, hyperuniform or not, and thus Anderson localization
can be present at all wavenumbers. Our results, combined with the methods to
generate media with a prescribed spectral density [24, 25, 26, 13], provide a
new inverse-design approach [56] to engineer and fabricate multilayered
dielectric media with novel wave properties.

Figure 1:  Schematic of three-dimensional disordered anisotropic stratified
media consisting of infinite parallel slabs of phases 1 (cyan) and 2 (dark
blue). A plane electromagnetic wave of transverse polarization is normally
incident into the medium, and its wavevector is shown as a red arrow.

##  2 Theory

###  2.1 Exact Strong-Contrast Expansion

Here, we briefly summarize the general nonlocal strong-contrast-expansion
formalism of the effective dynamic dielectric constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q},\omega\right)}}$
for 3D two-phase media with arbitrary symmetries [31]. (The strong-property-
fluctuation theory [57, 58] corresponds to a special case of our strong-
contrast formalism, as detailed in Sec. 1 of Supplement 1.) We consider a
macroscopically large two-phase composite specimen in three dimensions
embedded inside an infinitely large reference phase $q$ [59, 31]. For
simplicity, we take the phase $q$ to be the matrix phase (i.e., $q=1,2)$ and
assume that phases 1 and 2 are nonmagnetic and dielectrically isotropic with
real-valued and frequency-independent dielectric constants. These assumptions
imply the linear dispersion relation in the reference phase [i.e.,
$\mathinner{k_{q}\mathopen{\left(\omega\right)}}\equiv\absolutevalue{{\bf
k}_{q}(\omega)}=\sqrt{\varepsilon_{q}}\omega/c$], where $c$ is the speed of
light in vacuum, and thus we henceforth do not explicitly indicate the
$\omega$ dependence.

The general nonlocal strong-contrast expansion is a series expansion of the
linear fractional form of the tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q}\right)}}$, given
as

$\displaystyle\phi_{p}\bm{L}_{p}^{(q)}\cdot\quantity(\quantity{\bm{I}+\bm{D}^{(q)}\cdot\quantity[\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{q}\right)}}-{\varepsilon}_{q}\bm{I}]}\cdot\quantity[\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{q}\right)}}-{\varepsilon}_{q}\bm{I}]^{-1})\cdot\phi_{p}\bm{L}_{p}^{(q)}$
$\displaystyle=\phi_{p}\bm{L}_{p}^{(q)}-\sum_{n=2}^{\infty}\mathinner{\bm{\mathcal{A}}_{n}^{(p)}\mathopen{\left({\bf
k}_{q}\right)}},$ (1)

where $p(\neq q)$ indicates the polarized phase, $\bm{L}^{(q)}_{p}$ is the
expansion parameter defined as

$\displaystyle\bm{L}^{(q)}_{p}\equiv$
$\displaystyle\quantity({\varepsilon}_{p}-{\varepsilon}_{q})\quantity[\bm{I}+\bm{D}^{(q)}\quantity({\varepsilon}_{p}-{\varepsilon}_{q})]^{-1},$
(2)

and $\mathinner{\bm{\mathcal{A}}_{n}^{(p)}\mathopen{\left({\bf
k}_{q}\right)}}$ is a wavevector-dependent second-rank tensor that is a
functional involving the set of correlation functions
$S_{1}^{(p)},S_{2}^{(p)},\ldots,S_{n}^{(p)}$ and products of the principal
part of the dyadic Green’s function
$\mathinner{\bm{H}^{(q)}\mathopen{\left({\bf r}\right)}}$; see Sec. 1 of
Supplement 1. The series expansion (1) has four salient features. First, Eq.
(1) is derived from a spatially nonlocal averaged constitutive relation [60,
50, 61], resulting in an expansion that is valid from the long- to
intermediate-wavelength regimes. Second, the representation (1) exactly treats
multiple scattering to all orders at a given incident wavevector ${\bf k}_{q}$
when the nonlocal homogenization theory is valid because the terms
$\mathinner{\bm{\mathcal{A}}_{n}^{(p)}\mathopen{\left({\bf k}_{q}\right)}}$
for $n=2,\ldots$ in (1) explicitly account for complete microstructural
information (the infinite set of $S_{2},S_{3},\ldots$) to infinite order [31];
see Sec. 1 of Supplement 1 for details. Third, the choice of the shape of the
infinitesimal exclusion region (i.e., $\bm{D}^{(q)}$) leads to a different
expansion parameter $\bm{L}_{p}^{(q)}$ that determines the convergence
properties. Thus, unlike standard multiple-scattering theories [38, 62, 63],
here one can naturally ‘tune’ the general series expansion to obtain
distinctly different approximations suited for certain classes of
microstructures. Fourth, the left side of the series (1) is a linear
fractional transformation of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q}\right)}}$ rather
than $\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q}\right)}}$
itself, which leads to the rapid convergence of the strong-contrast expansions
so that its lower-order truncations approximate well higher-order functionals
(i.e., multiple scattering) of the exact series to all orders in terms of
lower-order diagrams [31].

###  2.2 Multiple-Scattering Approximations for Layered Media

We can now extract from the exact strong-contrast expansion (1) accurate
multiple-scattering approximations for layered media by truncating the
expansion at the $n$-point level. We focus here on such a formula at the two-
point level that depends on the spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}$ because it
is still accurate and easy to compute. For simplicity, we restrict ourselves
to normally incident waves (Fig. 1), and thus the effective dielectric
constant now depends on the wavenumber $k_{q}$. We outline the derivation here
and provide details in Sec. 2 of Supplement 1.

Since layered media have rotational symmetry about the $z$-axis and
translational symmetry in the $x-y$ plane, the spectral density can be
expressed as

$\displaystyle\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf
k}\right)}}=(2\pi)^{2}\mathinner{\delta\mathopen{\left(k_{x}\right)}}\mathinner{\delta\mathopen{\left(k_{y}\right)}}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k_{z}\right)}},$
(3)

where $\mathinner{\delta\mathopen{\left(k\right)}}$ is the one-dimensional
Dirac delta function, and
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k_{z}\right)}}$ is the
spectral density of 1D two-phase media. For 1D packings of identical hard rods
of radius $a$ and packing fraction $\phi_{2}$,
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k_{z}\right)}}=\phi_{2}[2\sin^{2}(k_{z}a)]/({k_{z}}^{2}a)\mathinner{S\mathopen{\left(k_{z}\right)}}$
[54, 64], where $\mathinner{S\mathopen{\left(k_{z}\right)}}$ is the structure
factor of the rod centers. Due to these symmetries, when applying the series
(1) to layered media, we utilize the feature 3 discussed in Sec. 2.1 by
choosing a disk-like exclusion region normal to the $z$-axis [31], leading to

$\displaystyle\bm{D}^{(q)}=$ $\displaystyle{\varepsilon_{q}}^{-1}\hat{{\bf
z}}\hat{{\bf
z}},\quad\bm{L}^{(q)}_{p}=\beta_{pq}\Big{[}\varepsilon_{p}\quantity(\bm{I}-\hat{{\bf
z}}\hat{{\bf z}})+\varepsilon_{q}\hat{{\bf z}}\hat{{\bf z}}\Big{]},$ (4)

where $\hat{{\bf z}}$ is a unit vector along the $z$-direction and
$\beta_{pq}$ is the one-dimensional counterpart of the _dielectric
polarizability_ , defined as $\beta_{pq}\equiv
1-\varepsilon_{q}/\varepsilon_{p}$. Here, $\bm{L}^{(q)}_{p}$ is obtained by
substituting $\bm{D}^{(q)}$ in (4) into (2). Using (4) and the assumption of
normal incidence, the general expression for
$\mathinner{\bm{\mathcal{A}}_{2}^{(p)}\mathopen{\left(k_{q}\right)}}$
simplifies as

$\displaystyle\mathinner{\bm{\mathcal{A}}_{2}^{(p)}\mathopen{\left(k_{q}\right)}}=$
$\displaystyle(\varepsilon_{p}\beta_{pq})^{2}\frac{\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k_{q}\right)}}}{\varepsilon_{q}}(\bm{I}-\hat{{\bf
z}}\hat{{\bf z}}),$ (5)
$\displaystyle\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k\right)}}=$
$\displaystyle\frac{{k}^{2}}{\pi}\mathrm{p.v.}\int_{0}^{\infty}\differential{q_{z}}\frac{\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{z}\right)}}}{{q_{z}}^{2}-{(2k)}^{2}}+\frac{ik}{4}[\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(0\right)}}+\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(2k\right)}}],$
(6)

where $\mathrm{p.v.}$ stands for the Cauchy principal value. Note that
$\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k_{q}\right)}}$ is the nonlocal
attenuation function for 1D two-phase media, whose 2D and 3D counterparts were
derived in Ref. [31].

Applying (4) and (5) into the second-order truncation of the series (1) yields

$\displaystyle\frac{(\phi_{p}\varepsilon_{p}\beta_{pq})^{2}}{\varepsilon_{q}(\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}/\varepsilon_{q}-1)}(\bm{I}-\hat{{\bf
z}}\hat{{\bf
z}})+\frac{(\phi_{p}\varepsilon_{q}\beta_{pq})^{2}}{\varepsilon_{q}(1-\varepsilon_{q}/\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{q}\right)}})}\hat{{\bf
z}}\hat{{\bf z}}$ $\displaystyle=$
$\displaystyle\varepsilon_{p}\beta_{pq}[\phi_{p}-(\varepsilon_{p}\beta_{pq})\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k_{q}\right)}}/\varepsilon_{q}](\bm{I}-\hat{{\bf
z}}\hat{{\bf z}})+\phi_{p}\varepsilon_{q}\beta_{pq}\hat{{\bf z}}\hat{{\bf
z}},$ (7)

where we have decomposed the effective dielectric constant tensor into two
orthogonal components
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}$ and
$\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{q}\right)}}$ for the
transverse and longitudinal polarizations, respectively, as follows:
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{q}\right)}}=\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}\quantity(\bm{I}-\hat{{\bf
z}}\hat{{\bf
z}})+\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{q}\right)}}\hat{{\bf
z}}\hat{{\bf z}}$. Now (7) provides two independent approximations:

$\displaystyle\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}=$
$\displaystyle\varepsilon_{q}\quantity[1+\frac{{\phi_{p}}^{2}(\varepsilon_{p}/\varepsilon_{q})\beta_{pq}}{\phi_{p}-(\varepsilon_{p}\beta_{pq})\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k_{q}\right)}}/\varepsilon_{q}}],$
(8)
$\displaystyle\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{q}\right)}}=$
$\displaystyle\varepsilon_{q}(1-\phi_{p}\beta_{pq})^{-1}.$ (9)

Note that $\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}$
is dependent on the incident wavenumber $k_{q}$ and is complex-valued if
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(0\right)}}+\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(2k_{q}\right)}}>0$,
implying that the media can be lossy due to forward scattering and
backscattering from inhomogeneities in the local dielectric constant. By
contrast, $\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{q}\right)}}$ is
independent of $k_{q}$, reflecting the fact that a traveling longitudinal wave
cannot exist under our assumptions. Hence, we focus on
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}$ in the rest
of this work. In the static limit, Eqs. (8) and (9) reduce to the arithmetic
and harmonic means of the dielectric constants, respectively:

$\displaystyle\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(0\right)}}=$
$\displaystyle\left\langle\varepsilon\right\rangle\equiv\phi_{p}\varepsilon_{p}+\phi_{q}\varepsilon_{q},$
$\displaystyle\mathinner{\varepsilon_{e}^{z}\mathopen{\left(0\right)}}=$
$\displaystyle(\phi_{p}/\varepsilon_{p}+\phi_{q}/\varepsilon_{q})^{-1}.$ (10)

Interestingly, these static results are exact for any microstructure [54].
Renormalization of the reference phase for the optimal convergence (Sec. 2 of
Supplement 1), equivalent to using the effective Green’s function in Ref.
[31], yields a _scaled_ strong-contrast approximation for disordered layered
media:

$\displaystyle\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}=\varepsilon_{q}\quantity[1+\frac{{\phi_{p}}^{2}(\varepsilon_{p}/\varepsilon_{q})\beta_{pq}}{\phi_{p}-(\varepsilon_{p}\beta_{pq})\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k_{q}\sqrt{\left\langle\varepsilon\right\rangle/\varepsilon_{q}}\right)}}/\left\langle\varepsilon\right\rangle}],$
(11)

where $\left\langle\varepsilon\right\rangle$ is given in (10). We henceforth
focus on this scaled approximation because it is more accurate than (8), as
shown in Fig. S6 in Supplement 1. As shown in Ref. [31], our formula (11)
satisfies the Kramers-Kronig relations [65] so that its predictions properly
exhibit both _normal dispersion_ [i.e., an increase in
$\real[\varepsilon_{e}^{\perp}]$ with $k_{q}$] and _anomalous dispersion_
[i.e., a decrease in $\real[\varepsilon_{e}^{\perp}]$ with $k_{q}$].
Furthermore, satisfying the Kramers-Kronig relations also implies that our
formula (11) yields qualitatively accurate predictions, even beyond the
validity regime (i.e., $k_{q}/\rho\lesssim 1.5$); see Sec. 5 and Sec. 6 on the
Supplement 1 for details.

In the scaled approximation (11), the quantity
$\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k_{q}\right)}}$ is generally
complex-valued at a given incident wavenumber $k_{q}$, producing a
corresponding $\varepsilon_{e}^{\perp}(k_{q})$ with a nonnegative imaginary
part. Following conventional usage, a composite attenuates waves at a given
wavenumber if the imaginary part of the effective dielectric constant is
positive. Such attenuation occurs here only because of multiple scattering
effects (not absorption).

Importantly, our strong-contrast approximation (11) predicts that stealthy
hyperuniform layered media can be perfectly transparent (i.e.,
$\imaginary[\varepsilon_{e}^{\perp}]=0$) in the infinite-volume limit for a
finite range of wavenumbers. Since any stealthy hyperuniform systems
completely suppress both forward scattering and backscattering [i.e.,
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(0\right)}}+\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(2k_{1}\right)}}=0$]
for $k_{1}<K/2$,
$\imaginary[\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k_{1}\right)}}]=0$,
and substituting this condition into the scaled formula (11) yields a perfect
transparency interval:

$0\leq
k_{1}<K_{T}\equiv\frac{K}{2\sqrt{\left\langle\varepsilon\right\rangle}}=\frac{\rho\pi\chi}{\sqrt{\left\langle\varepsilon\right\rangle}},$
(12)

where $\left\langle\varepsilon\right\rangle$ is the arithmetic mean of the
local dielectric constant; see (10). Perfect transparency implies an infinite
localization length within this spectral range, which is consistent with an
estimate that the localization length is inversely proportional to what we
call the spectral density [40]. We stress that the prediction (12) is purely
theoretical and hence does not rely on simulations or measurements of the
spectral density.

##  3 Model Microstructures

Figure 2:  Disordered stealthy hyperuniform layered media of packing fraction
$\phi_{2}=0.2$ and unit number density $\rho=1$ at three values of
$\chi=0.1,0.2,0.3$. (a) Representative images at (Top) $\chi=0.1$, (Middle)
$\chi=0.2$, and (Bottom) $\chi=0.3$. The particle phases are shown in colors.
(b) The spectral densities
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ as functions of
dimensionless wavenumber $k/\rho$.

Stealthy hyperuniform particle systems are defined by a spectral density that
vanishes in a finite range of wavenumbers that includes the origin
[$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}=0$ for
$0<\absolutevalue{{\bf k}}\leq K$]. The degree of stealthiness $\chi$ is
measured by the ratio of the number of the constrained wavenumbers in the
reciprocal space to the total degrees of freedom, i.e., in one dimension,
$\chi=K/(2\pi\rho)$, where $\rho$ is the number density of particles. For
$\chi<0.5$ in two and three dimensions or $\chi<1/3$ in one dimension, these
stealthy hyperuniform systems are highly degenerate and disordered [26]. Thus,
we consider 1D cases in which $\chi$ takes the following values:
$\chi=0.1,0.2,0.3$. Henceforth, we take the characteristic inhomogeneity
length scale $\xi$ to be the mean particle separation $1/\rho$, which is of
the order of the mean nearest-neighbor distance $\ell_{P}$, (i.e., $\xi\sim
1/\rho$). This choice means that the range of validity of our nonlocal theory
is $k_{1}/\rho\lesssim 1.5$ for the current models, as shown later in Sec. 5.

We numerically generate 1D packings of packing fraction $\phi_{2}=0.2$ in the
following two-step procedure. First, we generate point configurations of $N$
particles in a periodic fundamental cell $\mathfrak{F}$ via the _collective-
coordinate optimization_ technique [24, 25, 26], which numerically generates
ground states with very high-precision of the following potential energy:

$\mathinner{\Phi\mathopen{\left({\bf
r}^{N}\right)}}=\frac{1}{V_{\mathfrak{F}}}\sum_{\absolutevalue{{\bf
k}}<K}\mathinner{\tilde{v}\mathopen{\left({\bf
k}\right)}}\mathinner{S\mathopen{\left({\bf
k}\right)}}+\sum_{i<j}\mathinner{u\mathopen{\left(r_{ij}\right)}},$ (13)

where $\mathinner{\tilde{v}\mathopen{\left({\bf k}\right)}}=1$ for
$\absolutevalue{{\bf k}}<K$, $V_{\mathfrak{F}}$ is the volume of
$\mathfrak{F}$, and a soft-core repulsion term [66] is

$\mathinner{u\mathopen{\left(r\right)}}=\begin{cases}(1-r/\sigma)^{2},&r<\sigma,\\\
0,&\mathrm{otherwise}.\end{cases}$ (14)

The soft repulsion (14) is to get the stealthy hyperuniform ground states
whose nearest-neighbor distances are larger than the length scale $\sigma$
[16, 31]. Finally, we create packings with dielectric constant
$\varepsilon_{2}$ by circumscribing the points by identical rods of radius
$a<\sigma/2$ without overlaps [8]; see Fig. 2(a). The parameters used to
generate these systems are listed in Sec. 3 of Supplement 1.

We compute the spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ from the
generated packings as shown in Fig. 2(b). From the long- to intermediate-
wavelength regimes ($k/\rho\lesssim 10$), we clearly see that stealthy
hyperuniform packings exhibit a higher degree of correlations as $\chi$
increases. In the small-wavelength regime ($k/\rho\gg 10$ or, equivalently,
$ka\gg 1$), however, the curves tend to collapse onto a single curve,
reflecting the fact that these three systems consist of identical hard rods.

##  4 Simulations

We validate the accuracy of our predictions on the effective dielectric
constant $\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$
and the transmittance $T$ by comparing primarily to full-waveform simulations
[67] via an open-source FDTD package MEEP [68], although we use the transfer
matrix method to compute $T$, as explained in Supplement 1. We take the matrix
to be the reference phase (i.e., $q=1$) and the particles to be the polarized
phase (i.e., $p=2$) and set the phase contrast ratio as
$\varepsilon_{2}/\varepsilon_{1}=4$. We measure the transmittance spectra $T$
through the disordered stealthy hyperuniform layered media, which are then
compared to the predictions from (11). We also directly extract
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$ from the
nonlocal constitutive relation
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}=\left\langle\mathinner{\tilde{D}\mathopen{\left(k_{e},\omega\right)}}\right\rangle/\left\langle\mathinner{\tilde{E}\mathopen{\left(k_{e},\omega\right)}}\right\rangle$
at a given frequency $\omega$, as was done in Ref. [31]; see Fig. S6 of
Supplement 1. Here,
$\left\langle\mathinner{\tilde{D}\mathopen{\left(k_{e},\omega\right)}}\right\rangle$
and
$\left\langle\mathinner{\tilde{E}\mathopen{\left(k_{e},\omega\right)}}\right\rangle$
are the spatial Fourier transforms of the ensemble averages of dielectric
displacement field
$\left\langle\mathinner{D\mathopen{\left(x,\omega\right)}}\right\rangle$ and
electric field
$\left\langle\mathinner{E\mathopen{\left(x,\omega\right)}}\right\rangle$ at
the complex-valued effective wavenumber $k_{e}$, respectively; see details in
Sec. 3 of Supplement 1.

Figure 3: Real (upper) and imaginary (lower) parts of the nonlocal attenuation
functions $\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k\right)}}$ as
functions of dimensionless wavenumber $k/\rho$ for 1D disordered stealthy
hyperuniform layered media of packing fraction $\phi_{2}=0.2$ at three values
of $\chi=0.1,0.2,0.3$. Values are computed from the spectral densities in Fig.
2 and (6).

##  5 Results

Figure 4:  Predictions of the scaled strong-contrast approximation (11) of the
effective dynamic dielectric constant
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$ as a
function of the dimensionless incident wavenumber $k_{1}/\rho$ for disordered
stealthy hyperuniform layered media of $\phi_{2}=0.2$ and
$\varepsilon_{2}/\varepsilon_{1}=4$ at $\chi=0.1,0.2,0.3$. The lower panel is
a semilog plot of the imaginary part
$\imaginary[\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}]$.
For the effective dielectric constants, our theory is accurate up to
$k_{1}/\rho\lesssim 1.5$; see Fig. S6 of Supplement 1.

We now show how our multiple-scattering approximation (11) enables us to
accurately predict the real and imaginary parts of the effective dielectric
constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{1}\right)}}$ for disordered
stealthy hyperuniform layered media for $\chi=0.1,0.2,$ and $0.3$. We begin by
computing the nonlocal attenuation function
$\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k\right)}}$ given in (6) from
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$; see Fig. 3.
Stealthy hyperuniform layered media can achieve
$\imaginary[\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k\right)}}]=0$ up to a
finite wavenumber $k$, leading to the prediction of perfect transparency
interval; see (12).

Using the values of $\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k\right)}}$
in Fig. 3, we then compute the scaled approximation (11) for
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$; see Fig.
4. Both real and imaginary parts of these predictions show excellent agreement
with the results from FDTD simulations up to $k_{1}/\rho\lesssim 1.5$ (see
Fig. S6 of Supplement 1). In Fig. 4, the real part of our formula increases
with $k_{1}$ (normal dispersion) within the perfect transparency interval and
then decreases with $k_{1}$ (anomalous dispersion) outside of those intervals.
Such a spectral dependence of $\real[\varepsilon_{e}^{\perp}(k_{1})]$ comes
from the fact that the strong-contrast formula satisfies the Kramers-Kronig
relations [65] (see Sec. 6 of Supplement 1), as does the corresponding
approximation for 3D statistically isotropic media [31]. Our formula (11) also
shows qualitatively accurate dielectric responses even beyond the
intermediate-wavelength regime (i.e., $k_{1}/\rho\gtrsim 1.5$) because the
Kramers-Kronig relations closely relate the high-frequency predictions of
$\real[\varepsilon_{e}^{\perp}]$ to the accurate predictions of
$\imaginary[\varepsilon_{e}^{\perp}]$ in a finite spectral range and vice
versa [69].

Figure 5: Prediction (12) of the upper bound $K_{T}$ of the perfect
transparency intervals of 1D stealthy hyperuniform layered media with
$\phi_{2}=0.2$ and $\rho=1$ as a function of the phase contrast ratio
$\varepsilon_{2}/\varepsilon_{1}$. We consider three $\chi$ values:
$\chi=0.1,0.2,$ and $0.3$.

As shown in Fig. 4, these composites are perfectly transparent (i.e.,
$\imaginary[\varepsilon_{e}^{\perp}]=0$) for a wide range of frequencies, as
predicted by (12). At the edges of these transparency intervals, a
discontinuous change occurs in $\imaginary[\varepsilon_{e}^{\perp}]$ with
$k_{1}$ because the imaginary part is directly proportional to
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(2k_{1}\right)}}$; see
$\imaginary[\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k\right)}}]$ given in
(6). In higher dimensions, however, such abrupt transitions become
increasingly more difficult to achieve, as observed in Ref. [31]; see Sec. 2
of Supplement 1 for detail. Figure 5 depicts how the predicted perfect
transparency interval from (12) varies with the phase contrast ratio
$\varepsilon_{2}/\varepsilon_{1}$ for given values of $\chi$, $\phi_{2}$, and
$\rho$. We numerically demonstrate that these predicted intervals are valid
for $1<\varepsilon_{2}/\varepsilon_{1}<10$; see Fig. S3 of Supplement 1.

Figure 6:  Transmittance spectra $T$ as a function of the dimensionless
wavenumber $k_{1}/\rho$ for disordered stealthy hyperuniform layered media of
packing fraction $\phi_{2}\hskip 2.0pt(\equiv 2\rho a)=0.2$ and phase-contrast
ratio $\varepsilon_{2}/\varepsilon_{1}=4$ at three values of (a) $\chi=0.1$,
(b) $\chi=0.2$, and (c) $\chi=0.3$. The predictions are computed from (15)
using the scaled approximation (11). The green-shaded area indicates the
predicted transparency intervals (12). Within these intervals, our predictions
show excellent agreement with simulations, since there is no absorption.

From our scaled approximation (11), we also predict the _normal_ transmittance
$T$ through a layered medium at $k_{1}$ by assuming that the system is a
homogeneous slab of thickness $L$ with an effective dielectric constant
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$ and is
optically thin so that waves inside it can interfere coherently. To estimate
$T$, we use an Airy formula [46] of transmittance for a lossy homogeneous slab
with absorption:

$\displaystyle T$
$\displaystyle=\absolutevalue{\frac{-\sqrt{\varepsilon_{e}^{\perp}}\leavevmode\nobreak\
t^{2}\exp(i\sqrt{\varepsilon_{e}^{\perp}}k_{1}L)}{1-r^{2}\exp(2i\sqrt{\varepsilon_{e}^{\perp}}k_{1}L)}}^{2},$
(15)

where
$r\equiv(1-\sqrt{\varepsilon_{e}^{\perp}})/(1+\sqrt{\varepsilon_{e}^{\perp}})$
and $t\equiv 2/(1+\sqrt{\varepsilon_{e}^{\perp}})$, and
$\varepsilon_{e}^{\perp}$ is given by the approximation (11); see Sec. 4 of
Supplement 1 for results from (8). The electric field inside the dielectric
layered media attenuates solely due to multiple scattering, but its effective
behavior is identical to an exponentially damped wave due to absorption in a
lossy homogeneous medium (see Fig. S2 of Supplement 1). Hence, we expect our
formula (15) to provide a good approximation of $T$.

We now compare our theoretical predictions for transmission to the
corresponding results obtained from FDTD simulations; see Fig. 6. Remarkably,
our theory very accurately predicts the perfect transparency intervals (12)
i.e., no Anderson localization (green regions in Fig. 6), because they
correctly incorporate multiple scattering at finite wavelengths via the
spectral density. This observation is noteworthy because extended states in 1D
disordered systems are much more difficult to achieve than in higher
dimensions [38, 39, 41, 17]. Within these transparency intervals
($k_{1}<K_{T}$), our theory accurately predicts the small-amplitude periodic
oscillations in $T$ around unity, which come from coherent interference of the
multiply reflected waves due to the finite system thickness $L$ and thus
reduces to a constant close to unity when $L$ is much larger than the
coherence length of light [70, 46]. Outside of the perfect transparency
intervals, where scattering attenuation is strong, $T$ from the Airy formula
(15) provides a lower bound on the simulation results for the reason explained
in Sec. 4 of Supplement 1. However, we confirm that our theory still
accurately predicts, qualitatively, the spectral dependence of $T$ because
(11) yields a physically feasible dielectric response due to the Kramers-
Kronig relations. Within these strong-attenuation intervals, $T$ is
increasingly suppressed as $L$ increases and becomes virtually zero for
sufficiently large $L$ since
$\imaginary[\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}]>0$
if $k_{1}>K_{T}$. Stealthy hyperuniformity is required for disordered layered
media to possess perfect transparency for a finite range of wavenumbers.
Indeed, we show that such a perfect transparency interval cannot exist for
‘non-stealthy’ hyperuniform 1D media, even though these correlated disordered
systems anomalously suppress large-scale volume-fraction fluctuations; see the
example in Sec. 7 of Supplement 1.

##  6 Conclusions and Discussion

In summary, we have theoretically and numerically investigated the effective
wave properties, including the effective dynamic dielectric constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{q}\right)}}=\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}(\bm{I}-\hat{{\bf
z}}\hat{{\bf
z}})+\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{q}\right)}}\hat{{\bf
z}}\hat{{\bf z}}$ and transmittance $T$, of 3D statistically anisotropic two-
phase layered media made of 1D disordered stealthy hyperuniform packings. To
predict $\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{q}\right)}}$ of
such exotic disordered models, we derived for the first time a multiple-
scattering approximation (11) for statistically anisotropic media from the
strong-contrast-expansion formalism. Predictions of both the effective
dielectric constant
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{q}\right)}}$ and
transmittance $T$ are in excellent agreement with corresponding results
obtained from the FDTD simulations up to $k_{1}/\rho\lesssim 1.5$. Remarkably,
our predictions of $T$ are virtually identical to the simulation results
within the perfect transparency ranges. Our multiple-scattering approximation
is the first closed-form formula that provides a simple but accurate relation
between the effective wave properties of 3D layered media and their spectral
density that applies well beyond the quasistatic regime. Beyond the valid
range $(k_{1}/\rho\gtrsim 1.5)$, the formula (11) can still provide
qualitatively accurate and physically realistic predictions of dielectric
responses, since it satisfies the Kramers-Kronig relations.

We applied this newly derived formula (11) to disordered stealthy hyperuniform
layered media at $\chi=0.1,0.2$, and $0.3$. It is noteworthy that our
multiple-scattering formula (11) accurately predicts that these disordered
systems are perfectly transparent in the infinite-volume limit, implying no
Anderson localization, up to a finite wavenumber proportional to $\chi$ given
as (12). This observation is remarkable in that such extended states in 1D
disordered systems are more difficult to achieve than in higher dimensions
[38, 39, 40, 41, 17]. However, to further confirm that there is no Anderson
localization for the predicted perfect transparency range, it is necessary to
determine the localization length as a function of the sample size of these
stratified media theoretically and numerically, which is being carried out
elsewhere [71]. In contrast, for disordered non-stealthy layered media,
hyperuniform or not, our theory shows that there is no spectral range of
perfect transparency, implying that localization emerges as the system size
grows.

Our findings also have important practical implications. For example, we
clearly demonstrate that disordered stealthy hyperuniform layered media can be
employed as low-pass filters that transmit waves up to a selected wavenumber.
Furthermore, combining our theory with the capabilities to generate media with
a prescribed spectral density [24, 25, 26, 13] enables an inverse-design
approach [56] to engineer and fabricate layered dielectric materials with
novel wave properties. One possible design is a layered medium satisfying
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}=0$ for
$k<\epsilon$ and $2k_{L}<k<2k_{U}$, which transmits waves within a narrow
spectrum $k_{L}<k_{1}<k_{U}$. Notably, such computationally designed layered
media as well as other 1D disordered models can be readily fabricated via
vacuum deposition [72], spin-coating [73], and 3D printing techniques [74].
Thus, our results offer promising prospects for engineering novel
optoelectronic devices.

We note that our formalism can be extended to cases of obliquely incident
wavevectors ${\bf k}_{q}$, in which both effective dielectric constants
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left({\bf k}_{q}\right)}}$ and
$\mathinner{\varepsilon_{e}^{z}\mathopen{\left({\bf k}_{q}\right)}}$ become
important. Moreover, it also will be interesting to extend our theory to lossy
dielectric or metallic phases whose dielectric constants are now frequency-
dependent and complex-valued.

Funding The Research was sponsored by the Army Research Office and was
accomplished under Cooperative Agreement Number W911NF-22-2-0103.

Acknowledgments The authors thank Michael Klatt for helpful discussions.
Simulations were performed on computational resources managed and supported by
the Princeton Institute for Computational Science and Engineering (PICSciE).

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

Supplemental document See Supplement 1 for supporting content.

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###### Abstract

The letter submitted is an executive summary of our previous paper. To solve
the Einstein Podolsky Rosen “paradox” the two boundary quantum mechanics is
taken as self consistent interpretation of quantum dynamics. The difficulty
with this interpretation is to reconcile it with classical physics. To avoid
macroscopic backward causation two “corresponding transition rules” are
formulated which specify needed properties of macroscopic observations and
manipulations. The apparent classical causal decision tree requires to
understand the classically unchosen options. They are taken to occur with an
“incomplete knowledge” of the boundary states typically in macroscopic
considerations. The precise boundary conditions with given phases then select
the actual measured path and this selection is mistaken to happen at the time
of measurement. The apparent time direction of the decision tree originates in
an assumed relative proximity to the initial state. Only the far away final
state allows for classically distinct options to be selected from.
Cosmologically the picture could correspond to a big bang initial and a hugely
extended final state scenario. It is speculated that it might also hold for a
big bang/big crunch world. If this would be the case the Born probability
postulate could find a natural explanation if we coexist in the expanding and
the correlated CPT conjugate contracting world.

###### keywords:

two boundary interpretations of quantum mechanics; time symmetry; causality in
classical physics

157 10.3390/ecea-4-05010 2 Published: 20 November 2017 Causal Classical
Physics in Time Symmetric
Quantum Mechanics † Fritz W. Bopp Fritz W. Bopp Presented at the 4th
International Electronic Conference on Entropy and Its Applications, Basel,
Switzerland, 21 November–1 December 2017. Available online:
http://sciforum.net/conference/ecea-4. 03.65.Ta

What we know is Quantum Dynamics (called Bra-ket physics) establishing stable
states and describing matrix elements between an initial state and a final
state or analogous matrix elements in Quantum Field Theory at a given scale.
An unprecedented precision (sometimes with ten digits) makes the theory most
dependable.

Measurements, however, present a problem. Available concepts are not part of a
well tested theory. As pointed out a century ago by Einstein, Podolsky and
Rosen there is a paradox Einstein et al. (1935) which needs to be resolved (It
is by no means esoteric, p.e. Davidson (2017); Hellmuth et al. (1987)).
However this is not a problem of underlying Quantum Dynamics but of the
interface between Quantum Dynamics and Macroscopic Physics. Hence there are
two ways to resolve it. One can either amend or adjust Quantum Dynamics or
classical macroscopic concepts.

Considerable effort was invested in the first option. The Copenhagen
interpretation Bohr (1935) of Quantum Mechanics involving quantum jumps and a
limited ontology of waves of fields is reasonable but ugly. Penrose Penrose
(2005) is probably correct to speculate that the Hidden Variable Bell et al.
(1964); ’t Hooft (2016) and the Guiding Fields Broglie (1927); Bohm and
Aharonov (1957); Dürr and Teufel (2009) approaches do not work. New
experiments involving quantum interferences under gravitation Schlippert et
al. (2014) seem to exclude decoherence by gravitation Penrose (2005).
Deterministic approaches like the “Cellular Automaton” advocated by ’t Hooft
’t Hooft (2016) somehow lack motivation as they also lead to similar
complications as the ones discussed below.

We here search for a solution in the second approach. The paradigm Bopp (2017)
is to take the wave functions as real ontological objects following well
established quantum dynamics and to consider classical physics just as
effectively valid. Aspects of the causal macroscopic picture—we accepted a
long time ago as well known—might actually not be true on a fundamental level.
The huge body of observations could just be a good approximation somehow
reflecting our particular cosmological situation.

We do not see a way to describe the multi-faced evolution of the cosmos in a
meaningful mathematical even approximate framework. The picture presented
below is therefore largely limited to intuitive arguments. It develops needed
rules and somewhat daring concepts of required mechanisms envisioned to
recapture our macroscopic concepts.

With real ontological wave functions the Einstein Podolsky Rosen paradox means
instantaneous action over a large distance. Contrary to lore it does not
violate the essence of special relativity which just prescribes boost
transformations (Our hope is that following’t Hooft Hooft (2016) that general
relativity can be formulated in a unitary way even around singular structures
like black holes and that following Donoghue Donoghue and Menezes (2017) in
absense of of quantum jumps quantum field theory can be implemented in general
relativity involving integrals over products of fields. That coexisting paths
may lead to different final gravitational field configurations presents no
problem for the argument given below). However, using relativity instantaneous
action means backward causation in a different Lorentz system Argaman (2010)
which is widely considered unacceptable.

However it is not as bad as it looks. Quantum dynamics contains no time arrow
Sakurai and Napolitano (2014) and backward causation is not unacceptable.

The usual interpretation of quantum mechanics involves a fixed initial and
asymmetrically an open final state and quantum jumps. The original initial
state can, say, evolve to a state containing an electron with sidewards (say
right) spin. If a Stern Gerlach like measurement finds an upward spin the
original initial state is then replaced by the new one in a non unitary, non
time symmetric jump:

$<\mathrm{initial}\,|\,\mathrm{Measurement}\,|\to<\mathrm{initial}\,|\,\mathrm{Projection}\,|\,\,/\left|<\mathrm{initial}\,|\,\mathrm{Projection\,|}\,\right|$

The central point is that in a theory with backward causation the measurement
outcome like the up/down decision can just as well be done at a later time
then fixing the observed earlier observation.

$<\mathrm{initial}\,|\,U_{1}\,\mathrm{Projection}\,U_{2}\,|\leq\mathrm{initial}\,|\,U_{1}\,U_{2}\,\mathrm{Projection^{\prime}}\,|$

For this the measurement must involve sufficient witnesses so that the
projected state can be identified later on with a suitable new
Projection’-operator even if many projections are involved. It allows to apply
a two boundary picture where the decision is made by a final state encoding
all the measurement decisions. As for the jump formalism the unitary quantum
dynamics evolution has to be amended. One writes:

$<\mathrm{initial}\,|\,U_{1}\,\mathrm{Projection}\,U_{2}\,|\,\mathrm{final}>/<\mathrm{initial}\,|\,U_{1}\,\,U_{2}\,|\,\mathrm{final}>\,.$

to obtain a unit total probability. This two boundary formalism is well
developed by Aharonov and coworkers Aharonov et al. (1964) and others. We
stress that this is not a new theory which replaces the old one in a tricky
way which would require extensive work presumably of a group of people. The
formalism just uses the well established Quantum Dynamics extended it in a
straight forward way.

In contrast to the quantum world macroscopic considerations do not allow for
distinct coexisting path ways. A large number of effective measurements Joos
et al. (2013) must reduce ambiguities to allow for a macroscopic description.
In the two boundary description these measurements must stored in the final
boundary. This means that the overlap

$<\mathrm{inital}\,|\,\mathrm{final}>\sim 0.5^{\mathrm{decisions}}$

must be tiny which is possible. As also claimed by Aharonov and Cohen Aharonov
et al. (2017) it is a self consistent, time symmetric interpretation of
quantum theory.

The difficulty is to understand how the causal classical physics can arise in
such a frame work. To proceed we introduce two transition rules which prohibit
simple backward causation in classical physics.

The first one is known as no ‘post selection’ with macroscopic devices.
Consider a single photon state moving forward in a fiber. It is possible to
split the fiber in two and join it again with a macroscopically prearranged
relative phase. The forward going channel can be prohibited but this will not
affect the initial creation probability. As a consequence of unitarity other
channels (like reflection) have to open up.

The second rule states that states produced in a macroscopic distance have a
random relative phase. To explain the rule we consider a situation where it is
violated (for similar consideration see Alber et al. (2013)). Figure 1
considers two antennas in the focal points of an ellipsoid mirror. Within the
antennae clocked electronics allows to create preselected situations. With a
certain probability one emits a radio frequency photon at $-\Delta T$ which is
than absorbed at the other antenna at $+\Delta T$. If now both antennae emit
at $-\Delta T$ and absorb at $+\Delta T$ in a symmetric way the probability is
not effected. However if at $t=0$ the mirror gets dark on a point of positive
interference the initial emission probability at $-\Delta T$ is enhanced. This
second order interference effect actually constitutes a violation of the
second rule. The argument for the rule is that emissions with a synchronized
phases and absorptions not averaging out enhancements and depletions are
extremely rare in macroscopic situations and can therefore be ignored.

Figure 1: Second order interference.

In classical physics there is a causal decision tree. At each branching time a
decision how the future evolves is made. The critical point in the considered
framework is to understand the unchosen options. In a macroscopic
consideration the quantum phases are averaged out. With the resulting
“incomplete macroscopic knowledge” of the boundary states many path ways can
appear if the distance between the boundaries is sufficiently extended.

The apparent time direction of the decision tree originates in an assumed
relative proximity to the initial state (big bang). Given the initial and
present macroscopic state in all details there is even with incomplete
macroscopic knowledge only one path possible. It is not easy for
macroscopically different states with lots of witnesses to reach the same
final state. Only the assumed really distant final state allows for multiple
chosen and unchosen options.

If now the exact boundary conditions with their given phases are implemented
the actually taken path is determined. In a classical consideration this
selection is mistaken to happen at the instant of measurement. So it appears
that present decisions affects the choice of the future path.

The expanding universe is source of the thermodynamic arrow Zeh (2001). In a
closed box all intermediate histories will eventually reach a final state.
However, the hugely extended final state of our cosmos and significant
effectively interaction less regions make it plausible that all of today’s
macroscopic decisions can be encoded there. So the expansion is also source of
the effective time arrow considered here. Of course this is just a conjecture.
Many aspects of cosmology are not well known and it is not clear that in the
limit of a large final time the final state grows faster then needed for the
decision tree.

In a symmetric scenario with a big bang and a big crunch Gell-Mann and Hartle
(1994); Craig (1996) it might be enough to have an extremely extended
intermediate state at the turning point. If the initial and final state are
identical any selection collapses as all matching paths contribute and no
macroscopic aspects appear. If they are almost orthogonal both forward and
backward evolutions will produce two distinct extremely entangled intermediate
states. It is conceivable that considering the extreme extend of the
intermediate state the entanglement rarely matches and that effectively only
one intermediate state contributes.

The unfixed final state opens an amusing possibility. We consider the
situation with an electron wave the time $t$ in the forward moving world
($t<\frac{1}{2}T_{crunch}$) with spin in the rightward direction at and an
identical one at $T_{\mathrm{crunch}}-t$ in the opposite moving one. A
component $\left\langle\mathrm{rightward}\,|\,\mathrm{upward}\right\rangle$
leads to an upward intermediate state. We assume this state then to be
sufficiently traced in witnesses. The component which reaches the same
intermediate state in the backward moving world has the same magnitude
$\left\langle\mathrm{rightward}\,|\,\mathrm{upward}\right\rangle{}^{CPT}$.
Given the witnesses the common final state allows for no mixed contributions.
In both cases the remaining evolution should happen if a spin is chosen with
unity if the normalization for this case is taken into account. The
probability of an upward spin is therefore

$P(\mathrm{sideward}\to\mathrm{upward})=|\left\langle\mathrm{sideward}\,|\,\mathrm{upward}\right\rangle|^{2}$

and the Born rule is no longer a postulate but a consequence of the concept.

The seemingly statistical choice is no longer stored in a know-all final state
but in an intricate miss match of both “initial” states. The overlap is

$\displaystyle\left\langle bang\right|$ $\displaystyle U(T_{i},t)P_{up}\hskip
184.9429pt$ $\displaystyle U(t,T_{match})P_{match}U(T_{match},T_{crunch}-t)$
$\displaystyle\hskip 122.34692ptP_{up}U(T_{crunch}-t,T_{crunch})$
$\displaystyle\left|crunch\right\rangle$

for the upward measurement. A corresponding expression holds for the downward
one. We consider now for both cases the central second line for the considered
case of $P_{match}=1$. Their values are tiny, say $10^{-huge}$ resp.
$10^{-huge^{\prime}}$. With $50\%$ the value $huge$ is “würfelt” (Einstein’s
term for dice) larger. The natural statistical variation is of the order
$10^{-\sqrt{huge+huge^{\prime}}}$. Given its size this will always lead to an
exclusive dominance of one choice as the normalization needed in the two
boundary formalism is only applied after summation.

In such a scenario objects would exist with their wave function in the forward
moving world and with their conjugate function eons apart in tidily correlated
opposite moving one.

###### Acknowledgements.

We acknowledge useful comments of José M. Isidro and Giacomo D’ariano. The
authors declare no conflict of interest. References

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