Constraints on transmission, dispersion, and density of states in
  dielectric multilayers and stepwise potential barriers with arbitrary layer
  arrangement
Constraints on transmission, dispersion, and density of states
in dielectric multilayers and stepwise potential barriers with arbitrary layer arrangement
S. V. Zhukovsky1,2, ∗ and S. V. Gaponenko1, †
1Institute of Molecular and Atomic Physics, National Academy of Belarus, Nezavisimosti Ave. 70, 220072 Minsk, Belarus
2Physilakisches Institut, Universität Bonn, Nussallee 12, D-53115 Bonn, Germany
Normal-incidence transmission and dispersion properties of optical multilayers and one-dimensional stepwise
potential barriers in the non-tunneling regime are analytically investigated. The optical paths of every constituent
layer in a multilayer structure, as well as the parameters of every step of the stepwise potential barrier, are
constrained by a generalized quarter-wave condition. No other restrictions on the structure geometry is imposed,
i.e., the layers are arranged arbitrarily. We show that the density of states (DOS) spectra of the multilayer or
barrier in question are subject to integral conservation rules similar to the Barnett-Loudon sum rule but ocurring
within a finite frequency or energy interval. In the optical case, these frequency intervals are regular. For the
potential barriers, only non-periodic energy intervals can be present in the spectrum of any given structure, and
only if the parameters of constituent potential steps are properly chosen.
The integral conservation relations derived analytically have also been verified numerically. The relations can
be used in dispersion-engineered multilayer-based devices, e.g., ultrashort pulse compressors or ultracompact
optical delay lines, as well as to design multiple-quantum-well electronic heterostructures with engineered DOS.
I. INTRODUCTION
Over the centuries, the concept of homogeneity has played
a major part in both mathematics and physics. The very name
of a fundamental monograph on electrodynamics [1], Elec-
trodynamics of Continuous Media, suggests that there should
also exist electrodynamics of discontinuous media, quite dis-
tinct and yet unexplored. Indeed, most real-world physical
phenomena and processes are usually neither continuous nor
homogeneous, and all seemingly homogeneous substances are
in fact discontinuous on the molecular and atomic level. The
reason why the concept of homogeneous media is applicable
and produces good results in electrodynamics is that, in the
first place, the microscopic structure is so much smaller than
typical electromagnetic wavelengths that an effective-medium
approximation is valid. Secondly, many macroscopic systems
can be broken up into several homogeneous parts, the rela-
tively large size of which making the studies of the whole sys-
tem comparatively simple.
The intermediate case of mesoscopic structures where in-
homogeneities appear on the scale not minuscule enough to
use an effective-medium approach but not too large to allow
finite-size effects to be neglected has appeared more or less
recently. This was largely motivated by the advancement of
technology, allowing such structures to be fabricated and char-
acterized. Even the first steps in this direction have already
caused major advancements. The onset of semiconductor het-
erostructures was a breakthrough in electronics, the pioneers
in the area awarded the Nobel Prize in 2000 [2]. The intro-
duction of quantum mesoscopic systems such as nanocrystals
and quantum dots opens new horizons in many areas, includ-
ing biological sensor design and solid-state quantum compu-
tation (see, e.g., [3] and references therein). The introduction
∗Electronic address: sergei@th.physik.uni-bonn.de
†Electronic address: s.gaponenko@dragon.bas-net.by
of micro- and nanostructured optical materials has opened up
whole new areas of photonic crystal research, integrated op-
tics, and the newly-emerging metamaterial physics (see [4]),
with innumerable applications in telecommunication.
It appears that by arranging the matter in a mesoscopically
structured fashion, one can engineer its properties (e.g., elec-
tronic and/or optical) with considerable freedom. One can
achieve as rich a variety as seen among natural substances due
to a known diversity in their molecular-sized chemical compo-
sition. This freedom is especially increased when the concept
of structuring is extended beyond periodicity (see the recent
review [5]), such as in quasiperiodic [6] or fractal media [7].
Often it is even possible to design a structured medium in or-
der to achieve the chosen desired properties [8, 9]. Structured
media can even exhibit optical properties beyond what occurs
in natural materials, e.g., negative refraction [10] and the abil-
ity to slow down or stop light pulses [11].
All physical phenomena that involve interaction between
light and matter appear to be altered in inhomogeneous me-
dia. This alteration is believed to be a fundamental physical
principle involving modification of the properties of the vac-
uum (electromagnetic or electronic) in the vicinity of inhomo-
geneities. Such modification is generally described using the
concept of the density of states (DOS) [12]; however, this con-
cept is not without discussion points as regards definition of
the DOS in finite vs. infinite media [13]. Despite those diffi-
culties, the DOS concept appears a promising candidate for a
universal approach towards consistent description of physical
processes in arbitrarily inhomogeneous media.
As an example, it should be clear that the above men-
tioned modification of the vacuum cannot be totally arbitrary.
Causality had been shown to restrict the modification of spon-
taneous emission rate by spectral redistribution with the total
emission rate remaining unaffected (the Barnett-Loudon sum
rule [14]). Since spontaneous emission is related to the DOS,
this rule can be expressed as impossibility to change the total
“number” of states, but only to redistribute them spectrally,
which appears to be intuitively clear and heuristically potent.
In this paper we report on another, related limitation con-
http://arxiv.org/abs/0704.1897v2
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mailto:s.gaponenko@dragon.bas-net.by
cerning the modification of transmission and dispersion prop-
erties of optical dielectric multilayers as well as of electronic
heterostructures consisting of stepwise potential wells and
barriers in the non-tunneling regime.
For the optical case, making all constituent layers commen-
surate in optical path produces a set of equidistant single-
layer reflection-free (SLRF) points 2mω0 where (and only
where) the dispersion relation of the structure coincides with
that of a homogeneous medium (i.e., k(2mω0) ∝ 2mω0).
We have found that the optical DOS integrated between these
points does not depend on the structure geometry and does
not change if the constituent layers (whose optical paths are
commensurate with respect to each other) are rearranged. The
degree of modification to the optical properties as due to inho-
mogeneity of the structure is thus shown to be limited not only
in its amount but also in its extent. This means that not only is
a DOS enhancement in one spectral region compensated for
in some other region, but also the compensation must occur
within the distance 2ω0 between the SLRF points, which is
a spectral interval preset by the structural parameters of the
constituent element.
For the stepwise potential, similar relations have been
shown to exist. The single-layer reflection-free points do oc-
cur but are no longer equidistant. For any given values of pa-
rameters for constituent elements, there can be either none or
a multitude of aperiodically located SLRF points for all struc-
tures. In the latter case, the integral constraints on the DOS
can still be obtained, but they are more complex. In both op-
tical and quantum case, the analytical relations obtained have
been confirmed in numerical calculations.
The paper is organized as follows. In Sec. II we introduce
the reader to the structures under study and provide the neces-
sary basic notation. In Sec. III, we discuss the concept of the
DOS and its relation to the spectral properties of the structure.
In Secs. IV and V we derive the constraints on the DOS for
optical multilayers and for binary stepwise potential barriers,
respectively. In Sec. VI the results obtained in the previous
sections are compared and discussed. Finally, Sec. VII sum-
marizes the paper.
II. OPTICAL AND ELECTRONIC HETEROSTRUCTURES
We start by considering a one-dimensional dielectric mul-
tilayer nanostructure of N layers, each layer having a thick-
ness dj and a refractive index nj , infinite in the transverse
directions and surrounded on both sides by free space (n0 =
1). Consider a normally incident plane monochromatic wave
propagating through such a structure. This problem is one-
dimensional, and unless the multilayer structure contains op-
tically anisotropic materials, it can be described using scalar
electric field governed by the scalar Helmholtz equation [4]
E(x) + ε(x)
E(x) = 0. (1)
LetR(ω) and T (ω) denote the complex (i.e., taking into ac-
count the phase shift) reflection and transmission coefficients
of the multilayer structure, respectively. Let us now assume
that all the layers have such parameters that the optical path
njdj is the same for any j, so that
n1d1 = n2d2 = · · · = njdj = · · · = nNdN ≡ πc/2ω0. (2)
where ω0 is defined as the central frequency. We call any
structure that conforms to Eq. (2) the quarter wave (QW) mul-
tilayer structure.
It can be shown that for any even multiple of ω0 the prop-
agating wave passes each constituent layer without reflection
(no internal reflections at the layer interfaces), and thus gains
the phase shift ∆ϕ = ω
njdj , which is the same for all lay-
ers in view of the QW condition as expressed with Eq. (2).
(See also [15] for more detail on phase relations in Fresnel
reflection from one layer.) As a result, the structure becomes
fully transparent (|T (2mω0)| = 1) regardless of the number
or arrangement of constituent layers, and the total phase shift
becomes a simple sum of the shifts for all the layers:
T (ωm = 2mω0) = exp
 = exp (iNmπ) .
Eq. (3) essentially provides a set of equidistant frequency
points [we will call them single-layer reflection-free (SLRF)
points] where the propagation phase (and hence, the wave
number) is linearly dependent on frequency regardless of
the structure. Indeed, the dispersion relation at these points
j=1 dj)
km(ωm) = k(2mω0) =
linearly relates km and ωm, as is the case for a homogeneous
medium. This linear dependence occurs only at the set of
SLRF points ωm = 2mω0, and Eqs. (3)–(4) are not true any-
where between these points. Note that among all the transmis-
sion resonances present in a given multilayer’s spectrum, the
SLRF points represent stationary waves without any correla-
tions on a length scale greater than the optical path of one con-
stituent layer. As such, they are naturally the least localized
non-evanescent eigenstates possible in any given QW multi-
layer.
Moreover, QW structures are known to possess spectral pe-
riodicity in transmittance [16]
|T (ω + 2mω0)| = |T (ω)| (5)
and mirror symmetry within each period [16]
|T ((2m+ 1)ω0 + ω)| = |T ((2m+ 1)ω0 − ω)| ,
0 < ω < ω0.
Now let us note that Eq. (1) is isomorphic to the
Schrödinger equation governing a quantum particle with
mass mp and energy Ep in a stepwise potential u(x)
ψ(x) + [Ep − u(x)]ψ(x) = 0 (7)
Figure 1: A binary optical multilayer (a) together with its quantum mechanical counterpart: a particle with energy Ep = ~ω in 1D periodic
(b) and non-periodic (c) binary stepwise potential.
If the potential function is a constant (u(x) = −u0), the
solution of Eq. (7) is a plane-wave function
ψ0(x) = exp ikx = exp
Ep + u0
, (8)
which is analogous to a plane optical wave with the wave vec-
tor k =
2mp/~
Ep + u0. If Ep > u(x) for any x,
then k is real and the particle interacts with a potential barrier
in the non-tunneling (Ramsauer) regime. This is a quantum
mechanical analogy to electromagnetic wave propagation in a
dielectric structure. Similar to the optical case, one can con-
sider a stepwise potential barrier consisting of N “elementary
wells” (Fig. 1b,c). The role of refractive index is taken by the
potential energy uj in every step of the whole potential func-
tion. The frequency is replaced by the particle energy Ep,
which can be expressed in terms of de Broglie frequency [17]
as Ep = ~ω. The case Ep < u(x), which causes imaginary
wave vector in Eq. (8), is the tunneling case analogous to light
propagation in absorbing media (e.g., metals). It is outside
the scope of the present paper. To assure Ep > u(x), let us
assume Ep ≥ 0 and u(x) < 0 from now on.
Furthermore, it is commonly known that if the potential
represents a single step (u(x) = −u0 for, say, x < x0 and
u(x) = −u1 otherwise), one can introduce the coefficients
r01 =
k0 − k1
k0 + k1
, t01 =
k0 + k1
, (9)
which, when squared, denote the probability of finding the
impinging quantum particle reflected or transmitted, respec-
tively [17]. One can name them the reflection and transmis-
sion coefficient for matter waves, a potential step correspond-
ing to a single interface between dielectric media. Since the
expressions for r and t are the same (the wave vector k taking
the role of refractive index), one can use the same transfer-
matrix formalism for determining both the stationary electro-
magnetic wave distribution in a multilayer [18] and the steady-
state wave function for quantum particles travelling through a
complex stepwise potential [19]. Note that Eq. (8) indicates
that the “refractive index” introduced in this way possesses an
inherent quadratic dispersion.
Finally, we call a multilayer structure binary if it can be rep-
resented as consisting of two types of constituent layers (de-
noted as binary digits 0 and 1, following the notation in [20]),
to which two combinations of refractive index and thickness
(n0; d0) and (n1; d1) are attributed. By arranging the 0’s and
1’s in different sequences, it is possible to vary the geometry
of the structure very widely, making it periodic (if 0 and 1
alternate, as in 10101010101), disordered (if the sequence
is randomly determined), or deterministically aperiodic (e.g.,
quasiperiodic [6] or fractal [7]). A binary potential barrier,
with constituent elementary wells associated with (u0; d0) and
(u1; d1), can be introduced likewise. For brevity, we will oc-
casionally use the term “layers” for both types of constituent
elements.
Note that whenever the sequence contains two identical lay-
ers (e.g., “00” in 1010100101) , it will of course mean in prac-
tice that the corresponding structure will contain a single layer
with thickness 2d0. However, for the purpose of this work we
will regard such combinations as two separate constituent lay-
ers. The number of layers of both typesN0 andN1, as well as
their total number N = N0 + N1, will then remain the same
regardless of layer rearrangement, indicative of the transmis-
sion coefficient phase at the SLRF points [see Eq. (3)].
III. OPTICAL AND QUANTUM DENSITY OF STATES
As mentioned in Section I, any inhomogeneity present in
space is known to modify the properties of the quantum or
electromagnetic vacuum in its vicinity. This modification
takes the form of the change in the local DOS. It is believed
to affect all phenomena that involve light-matter interaction,
such as spontaneous emission or Raman scattering (see, e.g.,
[12] and references therein).
Physically, the local DOS N (r, E) is directly related to
the trace of Green’s function for the system in question:
N (r, E) ∝ Im Tr G(r, r, E). By taking the integral Green’s
functionG(E) in place of the local one, one obtains the value
of the DOS N (E) that is characteristic to the whole system for
a given value of energy. In a spatially finite system (a potential
well with infinite walls or a closed resonator), only the states
with a discrete set of energy (or frequency) eigenvalues are
allowed. Supposing that these eigenvalues are dense enough,
this integral DOS has a rather loose mathematically but very
intuitive meaning of the number of these discrete states per
unit energy.
In an open resonator, any value of energy corresponds to an
eigenstate, and the DOS transforms into a continuous spec-
trum N (E), indicative of spectral characteristics for the over-
Figure 2: (Color online) Transmittance |T (ω)|2 (dashed line) and normalized averaged local DOS ρ(ω)v0 [as of Eqs. (15) and (18), solid line]
for multilayer structures of different geometry: (a) single 9d1-thick layer; (b) 3-layer structure; (c) 9-layer periodic structure; (d) Fabry-Pérot-
like periodic structure with a half-wave defect; (e) coupled-defect structure; (f) fractal Cantor-like structure. All multilayers have N = 9. The
vertical scale is chosen alike for all plots for the ease of comparison, the insets showing the scale of clipped peaks. The area [0, 1] is shaded to
show the allowed region for transmittance, as well as to provide a visual guide for estimating the integral of ρ(ω) [see Eq. (19)].
all vacuum modification. It had been shown [21, 22, 23] that
a finite-sized inhomogeneous potential located in an infinite
1D space causes the local DOS integrated over the whole
space to undergo an overall finite modification ∆N (E) ≡
[N (x,E) −N0(x,E)] dx. Note that we are consider-
ing a finite-sized inhomogeneity in an infinite space (an open
resonator), as opposed to a finite system isolated from the out-
side space (a closed resonator). In the former, ∆N was found
to be proportional to the derivative of the total phase accumu-
lated by the wave packet during its transmission through the
inhomogeneity:
∆N (E) = (1/π) (dϕ/dE) , (10)
which, as seen from Eq. (8), becomes asymptotically zero for
very large energies compared to the potential (i.e., if nothing
gets in the particle’s way). A similar expression can be used
to determine the local density of electromagnetic states, also
called the optical DOS (for details on its definition in 2D and
3D case, see Refs. [13, 24]). The transition from local to in-
tegral DOS can be made in a similar manner to the quantum
system. In the 1D case (the wave propagation in a multilayer
is a 1D problem when only normal-incidence states are taken
into account) the modification to the optical DOS ∆N (ω) is
also likewise related to the derivative of transmission phase
[25]:
∆N (ω) = (1/π) (d(ϕ− ϕ0)/dω) . (11)
The subtraction of the free-space phase shift ϕ0 ensures
that ∆N (ω) = 0 in free space. In the work by Barnett and
Loudon [14] it has been shown that the modification of spon-
taneous emission rate Γ by inhomogeneous medium (as com-
pared to the free-space emission rate Γ0) integrated over the
whole spectrum must be zero (the Barnett-Loudon sum rule):
Γ(r, ω)
Γ0(ω)
dω = 0. (12)
The emission rate Γ is proportional to the local DOS N , and
the local optical DOS is frequency independent in free space
(see [12]). Spectral integration in Eq. (12) can be interchanged
with spatial integration over Γ(r, ω). Hence, a similar relation
holds for the integral DOS:
∆N (ω)dω = 0. (13)
The transition from local to integral DOS, as well as the
transition from∆N to N , involves renormalization and there-
fore may be ambiguous. A simple way to counter the diver-
gence is to accompany the transition from ϕ − ϕ0 to ϕ with
normalizing the DOS by the total thickness of the inhomoge-
neous medium D (see [26]):
ρ(ω) ≡ π∆N (ω)
, (14)
The authors in [26] simply define ρ as the optical DOS without
going into further details. We can see that it is in fact neither
local nor integral, but rather has the meaning of local DOS
modified by a finite inhomogeneity, averaged in infinite 1D
space. In the absence of any inhomogeneity, Eq. (14) gives
ρ(ω) = 1/c, a known value for the DOS in 1D free space.
Further, Eq. (14) can be used to calculate ρ(ω) from the
complex transmission coefficient T (ω) as
ρ(ω) =
[Im T (ω)]
Re T (ω)− Im T (ω) [Re T (ω)]′
D |T (ω)|2
, (15)
the derivation taken with respect to ω.
One must realize that the concept of the DOS introduced as
in Eqs. (10)–(11), and especially, as in Eq. (14), is not without
controversy. Questions arise already as to the physical mean-
ing of the quantities involved. For example, one can define a
“wave number” k a posteriori from the phase of the transmit-
ted wave
k(ω) =
Arg T (ω)
, (16)
which would equal the actual wave number in a homogeneous
medium, or the Bloch wave number in a periodic multilayer at
transmission resonances [26]. In such special cases, ρ would
equal the inverse group velocity (ρ(ω) = (dω/dk)−1), the
latter also equal to the energy velocity.
In the general case of non-periodic structures, however, the
concepts of phase, group, and energy velocity, as well as their
mutual correspondence, need to be re-examined. For instance,
the “phase time” defined as dϕ/dω is, in general, not equal to
the pulse’s actual “dwell time” (see [25, 27]), although, ad-
mittedly, both have a similar frequency dependence and in
some cases the phase time is a very good approximation for
the dwell time [28]. That said, it is safer not to assign any
direct physical meaning to k defined in Eq. (16) in the general
case. We will thus treat it like a parameter within the scope of
the present paper.
Another point is that the applicability of 1D models for
electromagnetic problems is in general of limited value. The
reason is that reduction of Maxwell’s equations to the scalar
wave equation (1), e.g., for multilayered media does not re-
ally make the problem entirely one-dimensional. In reality
one deals with finite-width beams rather than plane waves and
with excited atoms that can emit in any direction. The lat-
eral width of the multilayers is finite, too. As pointed out in
Ref. [29], the optical DOS reduces to the expression (14) only
if off-axis wave propagation is totally left out.
That kept in mind, the function ρ(ω) nevertheless under-
goes the same dramatic variation as does the transmittance
itself when the structure geometry is varied (see Fig. 2). The
peaks in |T (ω)|2 and ρ(ω) obviously correspond to each other.
Note that this correspondence is a physical property of multi-
layers rather than just a mathematical property of Eqs. (14)–
(15). Indeed, one can show analytically that ρ(ω) ∝ |T (ω)|2
for any single layer. This proportionality is due to the phase
structure of the Airy formulas, and is obviously not there for
arbitrary ϕ(ω) in Eq. (14). Numerical calculations confirm
that spectral features in |T (ω)|2 and ρ(ω) also correspond for
an N -layer structure, e.g., as seen in Fig. 2. It should be pos-
sible to show this analytically by induction but it is outside
the scope of the present paper. We note instead that the same
correspondence was observed in higher-dimensional systems
(e.g., in slab photonic crystals [30]).
Besides, one can observe that the sharper is the transmis-
sion resonance around some frequency ωr, the larger is the
value of ρ(ωr). Sharper transmission resonances correspond
to stationary waves with greater energy localization, and it
takes longer for greater energies to build up inside the struc-
ture. Hence, it takes longer for resonant transmission to mani-
fest in such cases. Therefore the maxima of ρ(ω) are just those
points where delayed light propagation is likely to be experi-
enced. The DOS spectrum is thus valuable as a quick visual
guide for determining the resonant behavior of any multilayer
structure, as employed earlier [31].
Finally, let us note that although k in Eq. (16) cannot be as-
signed a direct physical meaning in the general case, it can be
used as a parameter, which can provide some heuristic guid-
ance in experiments on the group velocity dispersion-related
effects (e.g., pertaining to propagation, compression, delay,
and chirp compensation of ultrashort laser pulses). For some
examples involving non-periodic structures, the reader is re-
ferred to Refs. [32, 33, 34].
IV. CONSTRAINTS IN MULTILAYERS
In the previous section, the use of ρ(ω) defined by Eq. (15)
as a meaningful characteristic of the structure’s optical prop-
erties has been motivated. It was demonstrated that ρ(ω) can
be strongly modified by altering the geometry of the structure
(Fig. 2). We proceed to show that the degree of geometry-
induced modification imposed on ρ has fundamental limita-
tions. One of these is the Barnett-Loudon sum rule – Eq. (13)
holds both for the quantum mechanical and for the electro-
magnetic case when the corresponding expression for ∆N (ω)
is used.
If the medium is a QW multilayer, the constraint becomes
stricter and involves integration over finite rather than infi-
nite frequency intervals. As the transmission properties in the
SLRF points are given by Eq. (3), the integral of ρ(ω) be-
tween those points can be evaluated explicitly using Eqs. (16)
Figure 3: (Color online) The schematic frequency dependence of propagation phase for (a) optical waves [Eq. (23)] and (b) quantum wave
function [Eq. (22)] in a slab of homogeneous dielectric and in a length of constant potential, respectively.
and (3)–(4) (see also [16]):
2(m+1)ω0
ρ(ω)dω =
k[2(m+1)ω0]
k[2mω0]
, (17)
which holds regardless of the geometrical arrangement of the
constituent layers in the structure, provided that the layers
obey the relation (2). One can further introduce the “bulk
velocity prameter” from the minimum time it takes light to
traverse the multilayer, internal reflections neglected, as
(djnj/c)
. (18)
This is a parameter independent either of ω or of the layer
arrangement of the structure. Making a transition to the di-
mensionless frequency η ≡ ω/ω0 and taking into account the
symmetry condition (6), we arrive at
v0ρ(η)dη = 1 (19)
and, further, since v0 = c and ρ0 = 1/c in free space,
∆ [v0ρ(η)] dη =
(v0ρ(η)− cρ0) dη = 0 (20)
for any integer m ≥ 0.
The conditions (19)–(20) have no less a universal charac-
ter than Eq. (13). They physically mean that the modification
of the transmission or dispersion properties due to layer rear-
rangement in QW multilayers is only possible within a finite
frequency range ω0. One can see in Fig. 2 that despite appar-
ently dramatic modification of ρ(ω), the enhancement in one
portion of the spectrum appears compensated by a gap in an-
other portion, so that the overall DOS, integrated between the
SLRF points, remains unaffected. It is also important to real-
ize that within ω0, one can achieve any desired spectral shape,
given the sufficient number of layers and sufficient freedom
in their arrangement. For example, a heuristic optimization
algorithm was recently used to demonstrate that certain ape-
riodic sequences can be employed to fabricate structures with
desired spectral properties [9].
Note, too, the inverse proportionality between ω0 and the
optical path of the constituent layers. It follows that if the
QW condition (2) is broken but the quantities njdj all remain
commensurate, the same reasoning can be applied. Eqs. (17)
and (19) can then be obtained by subdivision of the constituent
layers, accompanied by the according increase in the central
frequency (ω0 → Nω0). In the limiting case of mathemati-
cally incommensurate layers,N goes to infinity, and the struc-
ture appears to possess the same freedom as a continuously
inhomogeneous medium would, retaining only asymptotic re-
lation
ρ(ω)dω =
, (21)
associated with increasing ω0 to infinity in Eq. (20), and con-
sequently, representing the sum rule (13).
V. CONSTRAINTS IN POTENTIAL BARRIERS
The concept of optical DOS appeared in electrodynamics
largely by the influence of the quantum DOS. Such transfer
of concepts makes use of the analogy between the Helmholtz
(1) and Schrödinger (7) equations, as well as between a mul-
tilayer and a potential barrier, as outlined in Sec. II. In this
section, we will attempt to make these analogies work back-
wards and determine if, and to what extent, the relation (17)
can be generalized to the quantum mechanical case.
Consider a binary stepwise potential and the particle with
mass mp and energy Ep ≡ ~ω interacting with it in the non-
tunneling regime (Fig. 1b,c). Making use of the expression
(8) for k, we can derive the frequency dependence for prop-
agation phase of the particle’s wave function for constant po-
tential corresponding to one elementary potential well. Com-
pared to the same dependence for an optical wave in a homo-
geneous slab corresponding to one constituent layer, it has the
form (α ≡
2mp/~)
ϕ(qm)(ω) ≡ k(qm)(ω)d = dα
~ω + u; (22)
ϕ(opt)(ω) = ndω/c. (23)
Fig. 3 schematically shows both these dependencies. In the
optical case (Fig. 3a) the only variable parameter is the slope
given by nd. Hence, with the aid of Eq. (2) it becomes possi-
ble to achieve exactly the same dispersion relation, and hence
the same set of SLRF points, for both constituent layers when
n0 6= n1. This is what forms the foundation for reasoning pre-
sented in the previous section. In the quantum case (Fig. 3b)
u and d are seen to contribute in an essentially different way.
Is is thus not possible to arrive at the same dispersion relation
for two different potentials (u0 6= u1).
However, one can still define a set of frequency points
(though no longer equidistant) where ϕ(qm)(ωj) = jπ. In
these points, as can be seen from Eqs. (8) and (9), the whole
structure would be totally “transparent” for incoming quan-
tum particles (the Ramsauer effect). If the structure is binary,
the frequencies for both kinds of elementary wells are given
(0,1)
= j20,1π
2/d20,1α
2 − u0,1. (24)
Since two different parabolic curves can still have intersec-
tion points, one can manage to achieve ω
for two
pairs of j0 and j1. The reasoning presented in the previous
section can then be reproduced involving the quantity ρ de-
fined exactly as in Eqs. (14) and (17) and having the same
physical background. The dependence on ω, however, will be
more complex due to inherent dispersion as seen in Eq. (8).
For simplicity and for the sake of further analogy between
optical and quantum systems, let us require one of the equal
frequency pairs in Eq. (24) to correspond to ω = 0 (which is
always true for optical waves where all dispersion curves pass
trough the origin, see Fig. 3a). In this case we arrive at
α2d20u0 = α
2d21u1 = j
2, (25)
which can be seen as a quantum analogy to the condition (2).
The second pair (j0, j1) can then be found as an integer solu-
tion of the equation (first suggested in our earlier work [35])
j20 −
1− β/j2qw
j21 = β,
j0,1 > jqw; β ≡ (u0 − u1)α2d20/π2.
It can be seen that for any integer j0,1 > jqw there is a ratio-
nal β that solves Eq. (26). But β is related to the parameters
of the constituent potential wells. So, the inverse problem, i.e,
finding suitable j0,1 for a given β, is far more interesting from
a physical point of view. However, is not so straightforward
and is mathematically related to finding Pythagorean triples
in integer numbers. One can confirm numerically that there
are a multitude of solutions for many rational values of β (see
Table I). Some of them can be represented via recurrent rela-
tions, e.g., for β = 1 some of the solutions represent a series:
(i+1)
1 = j
1 + 6 + 4i, j
0 = j
1 − 1, (27)
where j
1 = j
(min)
qw = 2. Other cases are more complex,
but they, too, can be seen to form distinct solution branches
(Fig. 4).
Once j0 and j1 have been found, an analogous relation to
Eq. (17) can be formulated as
0=~ωjqw
ρ(E)dE =
[N0(j0 − jqw) +N1(j1 − jqw)] .
Note that Eq. (28) is more complicated than its optical
counterpart (17), and becomes, in general, dependent on the
number of constituent potential wells N1,2. This dependence
cannot be eliminated because one sees from Eq. (26) that it
is impossible to have j0 = j1 without violating the assertion
that j0,1 > jqw. It is still, however, completely independent of
layer rearrangement. In this sense, Eq. (26) represents a uni-
versal quantum mechanical conservation relation for the DOS
over a finite energy interval.
To demonstrate the results numerically, we have considered
a stepwise AlAs/GaAs quantum well (∆u = 1000 meV). To
aim at β = 4/5, we have taken d0 = 21.2 Å, d1 = 47.4 Å,
according to Eqs. (25)–(26). One possible solution of Eq. (26)
would then be jqw = 1, j0 = 2, j1 = 4 (see Table I). The
structures made of N = 9 elementary wells were used, and
the numbersN0 andN1 were fixed, too, at the values 4 and 5,
respectively.
We see in Fig. 5 that both ρ(E) and the transmission T (E)
are subject to quite a strong modification. It resembles the
modification seen in dielectric multilayers (compare, e.g.,
Fig. 5a,b with Fig. 2a,c). Two differences are the presence
of a decaying background due to the inherent dispersion [see
Eq. (22)] and the lack of periodicity because Eq. (2) can no
longer be satisfied.
However, if we integrate ρ(E) between the SLRF points
(~ωjqw = 0 and ~ωj0 = ~ωj1 = 3.75 eV) as provided by
Eq. (26), we can see that the integral does not change when
the layers are rearranged. Table II provides the results for nine
different structures and for several upper integration limits. It
can be seen that both below and above 3.75 eV the integrals
vary from structure to structure. When, however, the correct
integration limits are chosen, the difference vanishes and all
integrals equal 19, which is the right-hand side of Eq. (28) for
the chosen values of parameters.
VI. DISCUSSION
The equations (17)–(20) and (28) constitute the main result
of this paper pertaining to optical and electronic heterostruc-
tures, respectively. In both cases, we are dealing with conser-
vation of the DOS ρ integrated across a finite energy or fre-
quency region. As discussed in Sec. III, ρ represents the av-
eraged local DOS as modified by the presence of finite-sized
inhomogeneous structure in an infinite 1D free space. It is re-
lated to the dispersion and the transmission properties of the
heterostructures in question [see Eqs. (14)–(15)].
Table I: Some values of β that allow integer solutions of Eq. (26), along with some of such solutions obtained numerically.
β < 1 (jqw : j0, j1) β ≥ 1 (jqw : j0, j1)
1/4 (1 : 13, 15); (1 : 181, 209); (2 : 122, 126) 1 (2 : 7, 8); (2 : 26, 30); (3 : 17, 18); (3 : 99, 105); (4 : 31, 32)
1/3 (1 : 9, 11); (1 : 89, 109); (2 : 90, 94) 3/2 (2 : 8, 10); (2 : 68, 86); (3 : 63, 69)
1/2 (1 : 5, 7); (1 : 29, 41); (2 : 58, 62); (3 : 207, 213) 2 (2 : 10, 14); (2 : 58, 82); (3 : 45, 51)
2/5 (1 : 7, 9); (1 : 55, 71); (2 : 74, 78) 4 (3 : 7, 9); (3 : 18, 24); (4 : 14, 16); (4 : 52, 60); (5 : 23, 25)
2/3 (1 : 3, 5); (1 : 11, 19); (2 : 42, 46); (3 : 153, 159) 6 (3 : 9, 15); (3 : 33, 57); (4 : 16, 20)
4/5 (1 : 2, 4); (1 : 5, 11); (1 : 13, 19); (2 : 34, 38) 10 (4 : 8, 12); (4 : 32, 52); (5 : 35, 45)
Figure 4: The distribution of integer solutions j0,1 of Eq. (26) for (a) β = 1, (b) β = 4, and (c) β = 4/5. Distinct solution groups (“branches”)
can be seen.
These properties, as well as the DOS, can undergo dramatic
modification as compared to those of homogeneous media
(see Figs. 2 and 5) because a multilayer structure or a step-
wise potential barrier can be very complex. Nevertheless, the
modification appears to be limited both in its amount [see the
right hand side of Eqs. (17) and (28)] and in its extent (by the
finite integration limits in those equations).
There is an elegant physical explanation for the relations
obtained. By engineering the geometrical properties of an in-
homogeneous structure, it is only posible to redistribute the
Table II: Numerically evaluated integrals (D/~π)
ρ(E)dE [as in
Eq. (28)] from 0 to several upper energy values for nine structures
with N = 9, N0 = 4, and N1 = 5 (same as in Fig. 5). Standard
deviation of the values across all structures for each upper integra-
tion limit is provided in the lowest row. The limit of 3.75 eV (the
obtained value of the SLRF point) is accompanied by a drop in stan-
dard deviation down to 10−8, which falls within accuracy limits for
numerical integration.
Structure 0. . . 1 eV 3 eV 3.75 eV . . . 5 eV
001111100 7.5531 16.4036 19.0000 22.8568
010111100 7.5526 16.3991 19.0000 22.8593
100111100 7.5481 16.3982 19.0000 22.8582
110011100 7.5207 16.4016 19.0000 22.8574
010111010 7.5890 16.3991 19.0000 22.8600
100110011 7.5177 16.4048 19.0000 22.8592
100111001 7.5198 16.4017 19.0000 22.8512
110010011 7.5373 16.3996 19.0000 22.8603
101010101 7.5880 16.3982 19.0000 22.8657
Std. deviation 0.027 0.0045 8× 10−9 0.0038
available electromagnetic or quantum states across the spec-
trum, but impossible to alter the “total number” of the states.
The latter turns out to be related to the size or “1D volume” of
the structure [as seen by the presence of N at the right-hand
side of Eqs. (17) and (28)] and represents integrated character-
istics of the structure-affected vacuum. So, an enhancement of
the DOS in some pars of the spectrum (like the band edge res-
onances for a periodic structure in Fig. 2c) giving rise, e.g., to
the spontaneous emission enhancement, is inevitably accom-
panied by a suppression of the DOS in other spectral region
(like the band gap in the same figure), leading to the inhibition
of light propagation and all phenomena involving light-matter
interaction [4].
In this sense, the results obtained resemble already known
constraints on the DOS like the Barnett-Loudon sum rule (13).
However, in the relations obtained in this work the integra-
tion involved is finite rather than infinite. For the optical case,
this means a tighter restriction on the spectral redistribution of
the DOS. The compensation of suppression snd enhancement
must occur within the frequency interval ω0. This interval is
determined solely by the elementary constituent element of
the structure in question [see Eq. (2)]. It is totally indepen-
dent of geometrical arrangement of these elements. In other
words, the QW condition (2) enforces the existence of certain
points in the spectrum (the SLRF points) that cannot be “tran-
scended” by electromagnetic states that are “pushed around”
the spectrum by layer rearrangement.
On the other hand, the spectral properties of the structure
can be arbitrary everywhere between the SLRF points (3).
It should also be noted that the increase of N causes the
details in the spectra to become finer, and the variation of
T (ω) and ρ(ω) to get more rapid. These results can help to
understand the underlying physics of complex media.
Similar conclusions can be formulated for a quantum par-
Figure 5: The quantum averaged DOS ρ(E) (top) and transmittance
˛T (E)2
˛ (bottom) for an AlAs/GaAs quantum well and β = 4/5 in three
stepwise potential wells differing only by the elementary well rearrangement: (a) single-layer structure; (b) periodic structure; (c) non-periodic
structure. The portion between the SLRF points [0. . . 3.75 eV] is depicted, which corresponds to a solution of Eq. (26) for jqw = 1, j0 = 2,
j1 = 4.
ticle in a stepwise potential barrier. However, the inherent
quadratic dispersion as seen in Eqs. (8) and (22) results in
many differences. First and foremost, the SLRF points are
no longer guaranteed. Not only a relative restriction on con-
stituent elements (25) analogous to the QW condition (2) is
required, but also individual constraints on u0,1 and d0,1 are
necessary, so as to provide special values of β as determined
by Eq. (26). This makes the binarity of the structure an impor-
tant requirement in contrast to the optical case where Eqs. (2)
and (23) can be extended to as many kinds of constituent ele-
ments as needed. Because the equation (26) is quadratic rather
than linear, the SLRF points occur far more seldom than in the
optical case and are no longer equidistant. However, they still
do occur on a regular basis if they occur at all for a given
choice of parameters, as seen in Fig, 4. In this way, we have
provided a way for engineering an electronic heterostructure
where the DOS modification due to structure complexity is
confined in a finite spectral region. The structure itself can
be arbitrarily complex because Eqs. (24)–(26) do not depend
on N in any way.
To conclude this section, let us note that the structures in
question appear to possess other conservation relations. As
can be seen, e.g., in Fig. 2a–c, the transmission spectra contain
the same number of resonance peaks in the interval [0; 2ω0],
namely, nine, which equals the number of constituent layers.
Bearing a loose resemblance to the energy level splitting in
solids if one regards the layers as “atoms”, this was found
to be a general property of such multilayers [16]. However,
additional restrictions on the structures seem to be necessary,
such as the outermost layers of the structure being 1 rather
than 0 (compare, e.g., Figs. 2d, e). This requires additional
investigations and remains a subject for further studies.
VII. CONCLUSIONS AND OUTLOOK
To summarize, we have investigated the possible degree of
modification to transmission and dispersion properties, as well
as the averaged local DOS, in discretely inhomogeneous me-
dia. Both electromagnetic waves propagating in a dielectric
multilayer structure and a quantum particle propagating over
a stepwise, multiple-well potential barrier, have been consid-
ered (Fig. 1). In both cases, certain constraints on the con-
stituent elements of the structure [Eqs. (2) and (25)] allow to
derive the conservation relations over finite frequency or en-
ergy regions [Eqs. (17)–(20) and (28), respectively]. Both re-
lations hold regardless of the structure geometry (at least in
the sense of rearrangement of constituent elements) and are,
in this sense, universal, despite the fact that the spectral prop-
erties themselves can be strongly geometry-dependent. The
quantum case appears to be more complicated than the opti-
cal one and requires more conditions to be fulfilled, as im-
plied by a quadratic character of Eq. (26). The analytical re-
sults obtained have been verified by numerical calculations
(see Figs. 2, 5, and Table II).
The results obtained contribute to understanding the
physics of complex inhomogeneous media. They can be ap-
plied in the design of heterostructures with engineered disper-
sion, such as chirp compensation, pulse compression or delay
line devices. A more detailed studies of the relations obtained
would also be useful. It would be of interest to find out if, and
to what extent, the results can be applied to the case of optical
multilayers made of dispersive and/or absorptive materials, as
well as for potential barriers in the tunneling regime.
Acknowledgments
The authors are thankful to J. P. Dowling, H. V. Demir,
D. N. Chigrin, E. P. Petrov, and A. V. Lavrinenko for stimulat-
ing discussions, as well as to C. Kremers for helpful advice.
Partial support from the Basic Research Foundation of Be-
larus (Grant F03M-097) and the Deutsche Forschungsgemein-
schaft (Projects SPP1113 and FOR557), as well as the support
of the EC-funded projects PHOREMOST (FP6/2003/IST/2-
511616) is gratefully acknowledged.
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