text: The approximation scheme is based on equally spaced grid-points · · · x = 0 < x < < x = L, 0 1 N where − ∆x = x x = L/N. i+1 i System (37) is discretized using a central diﬀerence method − − − uN uN uN 2uN u˙N = uN 2 0 + κ 2 1 1 1 2∆−x ∆x2 − − uN uN uN + uN 2uN u˙N = uN 3 1 + κ 3 1 2 2 2 2∆x ∆x2 (39) . . . − − − uN uN− uN + uN− 2uN− u˙N− = uN− N N 2 + κ N N 2 N 1 N 1 N 1 2∆x ∆x2 (cid:2) ∈ (cid:3) where uN = uN = 0. For any v(x) C([0, L]), we deﬁne 0 N · · · ∈ − P N(v) = v(x ) v(x ) v(x − ) IRN 1 1 2 N 1 ∈ − For any vN IRN 1, deﬁne ∈ ΦN(vN) = v(x) C[0, L] · · · be the unique function of cubic spline determined by vN and (x , x , , x ) 0 1 N satisfying v(0) = v(L) = 0. We adopt L2-norm in C[0, L]. For any vector ∈ − vN IRN 1, its norm is deﬁned as (cid:88) − N 1 || || 2π vN 2 = v2 N N i i=1  (cid:12)  (cid:12) The space fo(cid:88)r es(cid:18)timation is deﬁned to be (cid:19) (cid:12)  (cid:12) (cid:88) (cid:12) ∈ W = α /2 + KF α cos(2kπ x) + β sin(2kπ x) (cid:12)(cid:12) αk, βk KIRF  0 k L k L α /2 + α = 0 0 k k=1 k=1 In this section, K = 2. This means that we want to ﬁnd the observability F for the ﬁrst ﬁve modes in the Fourier expansion of u(0). Or equivalently, we (cid:2) (cid:3) would like to ﬁnd the observability of α α β α β 0 1 1 2 2 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: approximation

text: scheme

text: is

text: based

text: on

text: equally

text: spaced

text: grid-points

text: x

text: x

text: x

text: L,

text: N

text: where

text: ∆x

text: x

text: x

text: L/N.

text: i

text: System

text: is

text: discretized

text: using

text: a

text: central

text: diﬀerence

text: method

text: uN

text: uN

text: uN

text: u˙N

text: uN

text: κ

text: 2∆−x

text: uN

text: uN

text: uN

text: uN

text: u˙N

text: uN

text: κ

text: uN

text: uN−

text: uN

text: uN−

text: u˙N−

text: uN−

text: N

text: N

text: κ

text: N

text: N

text: N

text: N

text: N

text: where

text: uN

text: uN

text: For

text: any

text: v(x)

text: L]),

text: we

text: deﬁne

text: N

text: P

text: N(v)

text: v(x

text: v(x

text: v(x

text: IRN

text: N

text: For

text: any

text: vN

text: IRN

text: deﬁne

text: ΦN(vN)

text: v(x)

text: L]

text: be

text: the

text: unique

text: function

text: of

text: cubic

text: spline

text: determined

text: by

text: vN

text: and

text: (x

text: x

text: x

text: N

text: satisfying

text: v(L)

text: We

text: adopt

text: in

text: L].

text: For

text: any

text: vector

text: vN

text: IRN

text: its

text: norm

text: is

text: deﬁned

text: as

text: N

text: 2π

text: vN

text: N

text: N

text: i

text: The

text: space

text: is

text: deﬁned

text: to

text: be

text: W

text: α

text: KF

text: α

text: x)

text: β

text: x)

text: αk,

text: βk

text: KIRF

text: k

text: L

text: k

text: L

text: α

text: α

text: k

text: In

text: this

text: section,

text: K

text: This

text: means

text: that

text: we

text: want

text: to

text: ﬁnd

text: the

text: observability

text: F

text: for

text: the

text: ﬁrst

text: ﬁve

text: modes

text: in

text: the

text: Fourier

text: expansion

text: of

text: Or

text: equivalently,

text: we

text: would

text: like

text: to

text: ﬁnd

text: the

text: observability

text: of

text: α

text: α

text: β

text: α

text: β

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 0.14 1.216e−1 0.12 0.1 0.08 min σ 0.06 0.04 3.820e−2 0.02 3.588e−3 3.994e−4 6.341e−5 4.810e−69.367e−7 0 1 2 3 4 5 6 7 8 Dimension Figure 1: Observability of heat equation of software and copyright issues are likely involved. For these types of ap- plications, a quantitative measure of partial observability is useful for several reasons. If a ﬁnite number of modes is enough to achieve accurate state approximates, guarantee the observability of these ﬁnite modes is a practi- cal solution. In large-scale networded systems with a very high dimension, operators may focus on a local area at a given period of time. In this case, the observability of the entire system is irrelevant. It is useful to achieve a partial observability just for the states directly related to the area of focus. For moving sensors, a quantitative measure of observability can be used as a cost function in ﬁnding optimal sensor locations in which the observability is maximized. Another issue to be addressed in this paper is consistency. In general the observability for PDEs is numerically computed using a system of ODEs as an approximation. However, it is not automatically guaranteed that the observability of the ODEs is consistent with the observability of the original PDE. In fact, a convergent discretization of a PDE may not preserve its observability. Take the following wave equation as an example − u u = 0, 0 < x < L, 0 < t < T tt xx u(0, t) = u(L, t) = 0, 0 < t < T (3) u(x, 0) = u (x), u (x, 0) = u (x), 0 < x < L 0 t 1 It is known that the total energy of the system can be estimated by using the energy concentrated on the boundary. However, in [20] it is proved that the discretized ODEs do not have the same observability. The energy of 5 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: min

text: σ

text: Dimension

text: Figure

text: Observability

label: caption

text: of

text: heat

text: equation

text: of

text: software

text: and

text: copyright

text: issues

text: are

text: likely

text: involved.

text: For

text: these

text: types

text: of

text: ap-

text: plications,

text: a

text: quantitative

text: measure

text: of

text: partial

text: observability

text: is

text: useful

text: for

text: several

text: reasons.

text: If

text: a

text: ﬁnite

text: number

text: of

text: modes

text: is

text: enough

text: to

text: achieve

text: accurate

text: state

text: approximates,

text: guarantee

text: the

text: observability

text: of

text: these

text: ﬁnite

text: modes

text: is

text: a

text: practi-

text: cal

text: solution.

text: In

text: large-scale

text: networded

text: systems

text: with

text: a

text: very

text: high

text: dimension,

text: operators

text: may

text: focus

text: on

text: a

text: local

text: area

text: at

text: a

text: given

text: period

text: of

text: time.

text: In

text: this

text: case,

text: the

text: observability

text: of

text: the

text: entire

text: system

text: is

text: irrelevant.

text: It

text: is

text: useful

text: to

text: achieve

text: a

text: partial

text: observability

text: just

text: for

text: the

text: states

text: directly

text: related

text: to

text: the

text: area

text: of

text: focus.

text: For

text: moving

text: sensors,

text: a

text: quantitative

text: measure

text: of

text: observability

text: can

text: be

text: used

text: as

text: a

text: cost

text: function

text: in

text: ﬁnding

text: optimal

text: sensor

text: locations

text: in

text: which

text: the

text: observability

text: is

text: maximized.

text: Another

text: issue

text: to

text: be

text: addressed

text: in

text: this

text: paper

text: is

text: consistency.

text: In

text: general

text: the

text: observability

text: for

text: PDEs

text: is

text: numerically

text: computed

text: using

text: a

text: system

text: of

text: ODEs

text: as

text: an

text: approximation.

text: However,

text: it

text: is

text: not

text: automatically

text: guaranteed

text: that

text: the

text: observability

text: of

text: the

text: ODEs

text: is

text: consistent

text: with

text: the

text: observability

text: of

text: the

text: original

text: PDE.

text: In

text: fact,

text: a

text: convergent

text: discretization

text: of

text: a

text: PDE

text: may

text: not

text: preserve

text: its

text: observability.

text: Take

text: the

text: following

text: wave

text: equation

text: as

text: an

text: example

text: u

text: u

text: x

text: L,

text: t

text: T

text: tt

text: xx

text: t)

text: u(L,

text: t)

text: t

text: T

text: u(x,

text: u

text: (x),

text: u

text: (x,

text: u

text: (x),

text: x

text: L

text: t

text: It

text: is

text: known

text: that

text: the

text: total

text: energy

text: of

text: the

text: system

text: can

text: be

text: estimated

text: by

text: using

text: the

text: energy

text: concentrated

text: on

text: the

text: boundary.

text: However,

text: in

text: it

text: is

text: proved

text: that

text: the

text: discretized

text: ODEs

text: do

text: not

text: have

text: the

text: same

text: observability.

text: The

text: energy

text: of

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: In this paper, we introduce a quantitative measure of partial observabil- ity for PDEs. Suﬃcient conditions are proved for the consistency of the observability for well-posed discretization schemes. 3 Partial Observability Consider a nonlinear initial value problem · · · × u = F (t, u, u , ), in Ω (0, T ] t x × { } u = u in Ω t = 0 (6) 0 H · y (t) = (u( , t)) u where Ω is an open set in IRn, F is a continuous function of t, u, and its derivatives with respect to x. Let X be a Banach space of functions deﬁned on Ω. The initial condition, u , lies in a subspace, V , of X. In · 0 0 the following, u(t) represents u( , t). A solution u(t) of (6), in either strict or weak sense, is a X-valued function in a subspace V of C0([0, T ], X). If additional boundary conditions are required, we assume that all functions in V satisfy the boundary conditions. We assume that (6) is locally well- posed in the Hadamard sense ([6, 17]) around a nominal trajectory. More speciﬁcally, let u(t) be an nominal trajectory. We assume that there is an ⊂ open neighborhood D V that contains u(0), such that • ∈ 0 0 For any u D , (6) has a solution. 0 0 • The solution is unique. • The solution depends continuously on its initial value. Proving the well-posedness of nonlinear PDEs is not easy. Nevertheless, for well-posed problems the consistency of observability is guaranteed. H · In (6), (u( , t)) or in short notation y (t), represents the output of the u H system associated with the solution u(x, t), where is a mapping, linear or · nonlinear, from X to IRp. We assume that y ( ) stays in a normed space of u || · || H functions from [t , t ] to IRp. Its norm is denoted by . We say that 0 1 Y { }∞ is continuous in a subset of C0([0, T ], X) if for any sequence u (t) in ∈ k k=k0 the subset and function u(t) V , → || − || uk(t) u(t) uniformly on [0, T ] implies kl→im∞ yuk yu Y = 0 (7) Instead of the entire state space, the observability is deﬁned in a ﬁnite dimensional subspace. Let { · · · } W = span e , e , , e 1 2 s 7

text: In

text: this

text: paper,

text: we

text: introduce

text: a

text: quantitative

text: measure

text: of

text: partial

text: observabil-

text: ity

text: for

text: PDEs.

text: Suﬃcient

text: conditions

text: are

text: proved

text: for

text: the

text: consistency

text: of

text: the

text: observability

text: for

text: well-posed

text: discretization

text: schemes.

label: section

text: Partial

text: Observability

text: Consider

text: a

text: nonlinear

text: initial

text: value

text: problem

text: u

text: F

text: (t,

text: u,

text: u

text: in

text: Ω

text: T

text: t

text: x

text: u

text: u

text: in

text: Ω

text: t

text: H

text: y

text: (t)

text: (u(

text: t))

text: u

text: where

text: Ω

text: is

text: an

text: open

text: set

text: in

text: IRn,

text: F

text: is

text: a

text: continuous

text: function

text: of

text: t,

text: u,

text: and

text: its

text: derivatives

text: with

text: respect

text: to

text: x.

text: Let

text: X

text: be

text: a

text: Banach

text: space

text: of

text: functions

text: deﬁned

text: on

text: Ω.

text: The

text: initial

text: condition,

text: u

text: lies

text: in

text: a

text: subspace,

text: V

text: of

text: X.

text: In

text: the

text: following,

text: u(t)

text: represents

text: u(

text: t).

text: A

text: solution

text: u(t)

text: of

text: in

text: either

text: strict

text: or

text: weak

text: sense,

text: is

text: a

text: X-valued

text: function

text: in

text: a

text: subspace

text: V

text: of

text: T

text: X).

text: If

text: additional

text: boundary

text: conditions

text: are

text: required,

text: we

text: assume

text: that

text: all

text: functions

text: in

text: V

text: satisfy

text: the

text: boundary

text: conditions.

text: We

text: assume

text: that

text: is

text: locally

text: well-

text: posed

text: in

text: the

text: Hadamard

text: sense

text: around

text: a

text: nominal

text: trajectory.

text: More

text: speciﬁcally,

text: let

text: u(t)

text: be

text: an

text: nominal

text: trajectory.

text: We

text: assume

text: that

text: there

text: is

text: an

text: open

text: neighborhood

text: D

text: V

text: that

text: contains

text: such

text: that

text: For

text: any

text: u

text: D

text: has

text: a

text: solution.

text: The

text: solution

text: is

text: unique.

text: The

text: solution

text: depends

text: continuously

text: on

text: its

text: initial

text: value.

text: Proving

text: the

text: well-posedness

text: of

text: nonlinear

text: PDEs

text: is

text: not

text: easy.

text: Nevertheless,

text: for

text: well-posed

text: problems

text: the

text: consistency

text: of

text: observability

text: is

text: guaranteed.

text: H

text: In

text: (u(

text: t))

text: or

text: in

text: short

text: notation

text: y

text: (t),

text: represents

text: the

text: output

text: of

text: the

text: u

text: H

text: system

text: associated

text: with

text: the

text: solution

text: u(x,

text: t),

text: where

text: is

text: a

text: mapping,

text: linear

text: or

text: nonlinear,

text: from

text: X

text: to

text: IRp.

text: We

text: assume

text: that

text: y

text: stays

text: in

text: a

text: normed

text: space

text: of

text: u

text: H

text: functions

text: from

text: [t

text: t

text: to

text: IRp.

text: Its

text: norm

text: is

text: denoted

text: by

text: We

text: say

text: that

text: Y

text: is

text: continuous

text: in

text: a

text: subset

text: of

text: T

text: X)

text: if

text: for

text: any

text: sequence

text: u

text: (t)

text: in

text: k

text: the

text: subset

text: and

text: function

text: u(t)

text: V

text: uk(t)

text: u(t)

text: uniformly

text: on

text: T

text: implies

text: kl→im∞

text: yuk

text: yu

text: Y

text: Instead

text: of

text: the

text: entire

text: state

text: space,

text: the

text: observability

text: is

text: deﬁned

text: in

text: a

text: ﬁnite

text: dimensional

text: subspace.

text: Let

text: W

text: span

text: e

text: e

text: e

text: s


text: (cid:84) where uˆ satisﬁes · · · uˆ = F (t, uˆ, uˆ , ) t ∈ x uˆ(0) W D (11) || − ||0 uˆ(0) u(0) = ρ X Then ρ/(cid:15) is called the unobservability index of u(0) along the trajectory u(t) at the level of ρ. Remark. The ratio ρ/(cid:15) can be interpreted as follows: if the maximum er- ror of the measured output, or sensor error, is (cid:15), then the worst estimation error of u(0) is ρ. Therefore, a small value of ρ/(cid:15) implies strong observ- ability of u(0). Diﬀerent from most traditional deﬁnitions of observability, W is a subspace of the state space. However, in the special case that W equals the entire state space that has a ﬁnite dimension and if the system is linear, (cid:15)2/ρ2 equals the smallest eigenvalue of the observability Gramiam (See Section ). Remark. The eﬀectiveness of this concept is veriﬁed in [10] using Burg- ers’ equation. A 4D-Var data assimilation, a method of state estimation in weather prediction, is applied to several data sets from diﬀerent sensor lo- cations. The result shows that the estimation using sensor locations with a smaller unobservability index results in more accurate estimates than those using the data from other sensor locations with higher unobservability in- dices. To numerically compute a system’s observability, (6) is approximated using ODEs. In this paper, we consider a general approximation scheme using a sequence of ODEs, ∈ duN/dt = F N(t, uN), uN IRN (12) uN(0) = uN 0 and two sequences of linear mappings → P N : V IRN 0 → (13) ΦN : IRN X ||·|| The norm in IRN is represented by . We assume that the approximation N scheme is well-posed, more speciﬁcally • For any bounded set B in D , there exist M > 0 and α > 0 so that 0∈ any solution of (6) with u(0) B satisﬁes || − || ≤ M u(t) ΦN(uN(t)) (14) X Nα 9 ##LTLine##

text: where

text: uˆ

text: satisﬁes

text: uˆ

text: F

text: (t,

text: uˆ,

text: uˆ

text: t

text: x

text: W

text: D

text: ρ

text: X

text: Then

text: is

text: called

text: the

text: unobservability

text: index

text: of

text: along

text: the

text: trajectory

text: u(t)

text: at

text: the

text: level

text: of

text: ρ.

text: Remark.

text: The

text: ratio

text: can

text: be

text: interpreted

text: as

text: follows:

text: if

text: the

text: maximum

text: er-

text: ror

text: of

text: the

text: measured

text: output,

text: or

text: sensor

text: error,

text: is

text: then

text: the

text: worst

text: estimation

text: error

text: of

text: is

text: ρ.

text: Therefore,

text: a

text: small

text: value

text: of

text: implies

text: strong

text: observ-

text: ability

text: of

text: Diﬀerent

text: from

text: most

text: traditional

text: deﬁnitions

text: of

text: observability,

text: W

text: is

text: a

text: subspace

text: of

text: the

text: state

text: space.

text: However,

text: in

text: the

text: special

text: case

text: that

text: W

text: equals

text: the

text: entire

text: state

text: space

text: that

text: has

text: a

text: ﬁnite

text: dimension

text: and

text: if

text: the

text: system

text: is

text: linear,

text: equals

text: the

text: smallest

text: eigenvalue

text: of

text: the

text: observability

text: Gramiam

text: (See

text: Section

text: Remark.

text: The

text: eﬀectiveness

text: of

text: this

text: concept

text: is

text: veriﬁed

text: in

text: using

text: Burg-

text: ers’

text: equation.

text: A

text: data

text: assimilation,

text: a

text: method

text: of

text: state

text: estimation

text: in

text: weather

text: prediction,

text: is

text: applied

text: to

text: several

text: data

text: sets

text: from

text: diﬀerent

text: sensor

text: lo-

text: cations.

text: The

text: result

text: shows

text: that

text: the

text: estimation

text: using

text: sensor

text: locations

text: with

text: a

text: smaller

text: unobservability

text: index

text: results

text: in

text: more

text: accurate

text: estimates

text: than

text: those

text: using

text: the

text: data

text: from

text: other

text: sensor

text: locations

text: with

text: higher

text: unobservability

text: in-

text: dices.

text: To

text: numerically

text: compute

text: a

text: system’s

text: observability,

text: is

text: approximated

text: using

text: ODEs.

text: In

text: this

text: paper,

text: we

text: consider

text: a

text: general

text: approximation

text: scheme

text: using

text: a

text: sequence

text: of

text: ODEs,

text: duN/dt

text: F

text: N(t,

text: uN),

text: uN

text: IRN

text: uN

text: and

text: two

text: sequences

text: of

text: linear

text: mappings

text: P

text: N

text: V

text: IRN

text: ΦN

text: IRN

text: X

text: The

text: norm

text: in

text: IRN

text: is

text: represented

text: by

text: We

text: assume

text: that

text: the

text: approximation

text: N

text: scheme

text: is

text: well-posed,

text: more

text: speciﬁcally

text: For

text: any

text: bounded

text: set

text: B

text: in

text: D

text: there

text: exist

text: M

text: and

text: α

text: so

text: that

text: 0∈

text: any

text: solution

text: of

text: with

text: B

text: satisﬁes

text: M

text: u(t)

text: ΦN(uN(t))

text: X

text: Nα

text: ##LTLine##


text: IEEE ANTENNA AND WIRELESS PROPAGATION LETTERS 2013 1 Unveiling Magnetic Dipole Radiation in Phase-Reversal Leaky-Wave Antennas Shulabh Gupta, Member, IEEE, Li Jun Jiang, Senior Member, IEEE, and Christophe Caloz, Fellow, IEEE Abstract— The radiation principle of travelling-wave type scanning. This conﬁguration uses periodic phase-reversal ra- phase-reversal antennas is explained in details, unveiling the diating elements interconnected by balanced transmission line presence of magnetic-dipole radiation in addition to well-known sections [10]. electric dipole radiation. It is point out that such magnetic- dipole radiation is speciﬁc to the case of traveling-wave phase- reversal antennas whereas only electric-dipole radiation exists in c b resonant-type phase-reversal antennas. It is shown that a phase- g + reversal travelling-wave antenna alternately operates as an array d - e of magnetic dipoles and an array of electric-dipoles during a time-harmonic period. This radiation mechanism is conﬁrmed + x y a through both full-wave and experimental results. - Index Terms— Balanced transmission line, full-space scanning, + g leaky-wave antenna, phase-reversal, magneto-electric antennas. - (a) I. INTRODUCTION c b + Periodic leaky-wave structures are a versatile class of - travelling-wave antennas capable of frequency-controlled x x beam steering, offering high directivity and simple feeding + a - mechanism [1]. With the advent of composite right/left-handed (CRLH) leaky-wave antennas capable of full-space scanning, + from backward to forward including broadside directions, - there has been a renewed interest in such structures [2]. (b) Full-scpae scanning was made possible by suppressing the c b open stopband and equalizing the impedance around broad- + side [2][3][4]. Following these advances, periodic leaky-wave - - antennas have become most attractive as low-cost alternative to phased array antennas [5]. + a - One approach to achieve a full-space beam scanning in leaky-wave structures, is to use the phase-reversal technique. + + ◦ x x This technique involves introducing a 180 phase shift along - the axis of the structure, so that λ /2-spaced periodically (c) g elements get excited in phase and therefore radiate. This tech- Fig. 1. Principle of phase-reversal antenna explained in terms of current nique is common in slot-waveguide antenna arrays where the ◦ distributions in two adjacent unit cells. a) Differential transmission line required extra 180 phase-shift is induced by proper location without phase reversal. b) Phase reversed differential lines at t = 0 and, c) at of the slots along the waveguide aperture [6]. A transmission- t = T/4, where T = 1/f0 is the time period of an assumed time-harmonic line counterpart of this structure is the Sterba-curtain antenna, signal of frequency f0. which utilizes periodic transmission-line cross-overs, placed λ /2 apart, to produce electric-dipole radiation from the cross- While the radiation mechanism of traditional transmission- g overs [7]. However, these waveguide and transmission-line line type phase-reversal antennas is well understood in reso- antennas were resonant in nature and therefore featured a nant conﬁgurations and extensively discussed in the literature small impedance-matching bandwidth product [8],[9]. This [11], travelling-wave phase-reversal antennas have not been phase reversal concept was later extended to a travelling- exhaustively explored. Speciﬁcally, they have been described wave conﬁgurations, thereby both improving the impedance- in terms of electric-dipole radiation only [10]. In this work, matching bandwidth performance and providing full-space the radiation mechanism of a travelling-wave type phase reversal antenna is revisited in details and unveils the presence S. Gupta and L. J. Jiang are with the Electrical and Electronic Engineering of magnetic-dipole radiation operating in conjunction with Department of University of Hong Kong, China (SAR). electric-dipole radiation. This dual radiation mechanism is C. Caloz is with the Department of Electrical Engineering, PolyGrames Research Center, E´cole Polytechnique de Montre´al, Montre´al, QC, Canada. speciﬁc to travelling-wave type phase-reversal antennas and ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: IEEE

text: ANTENNA

text: AND

text: WIRELESS

text: PROPAGATION

text: LETTERS

text: Unveiling

label: title

text: Magnetic

text: Dipole

text: Radiation

text: in

text: Phase-Reversal

text: Leaky-Wave

text: Antennas

text: Shulabh

label: author

text: Gupta,

text: Member,

text: IEEE,

text: Li

text: Jun

text: Jiang,

text: Senior

text: Member,

text: IEEE,

text: and

text: Christophe

text: Caloz,

text: Fellow,

text: IEEE

text: Abstract—

text: The

label: abstract

text: radiation

text: principle

text: of

text: travelling-wave

text: type

text: scanning.

text: This

text: conﬁguration

text: uses

text: periodic

text: phase-reversal

text: ra-

text: phase-reversal

text: antennas

text: is

text: explained

text: in

text: details,

text: unveiling

text: the

text: diating

text: elements

text: interconnected

text: by

text: balanced

text: transmission

text: line

text: presence

text: of

text: magnetic-dipole

text: radiation

text: in

text: addition

text: to

text: well-known

text: sections

text: electric

text: dipole

text: radiation.

text: It

text: is

text: point

text: out

text: that

text: such

text: magnetic-

text: dipole

text: radiation

text: is

text: speciﬁc

text: to

text: the

text: case

text: of

text: traveling-wave

text: phase-

text: reversal

text: antennas

text: whereas

text: only

text: electric-dipole

text: radiation

text: exists

text: in

text: c

text: b

text: resonant-type

text: phase-reversal

text: antennas.

text: It

text: is

text: shown

text: that

text: a

text: phase-

text: g

text: reversal

text: travelling-wave

text: antenna

text: alternately

text: operates

text: as

text: an

text: array

text: d

text: e

text: of

text: magnetic

text: dipoles

text: and

text: an

text: array

text: of

text: electric-dipoles

text: during

text: a

text: time-harmonic

text: period.

text: This

text: radiation

text: mechanism

text: is

text: conﬁrmed

text: x

text: y

text: a

text: through

text: both

text: full-wave

text: and

text: experimental

text: results.

text: Index

text: Terms—

text: Balanced

text: transmission

text: line,

text: full-space

text: scanning,

text: g

text: leaky-wave

text: antenna,

text: phase-reversal,

text: magneto-electric

text: antennas.

text: (a)

text: I.

text: INTRODUCTION

text: c

text: b

text: Periodic

text: leaky-wave

text: structures

text: are

text: a

text: versatile

text: class

text: of

text: travelling-wave

text: antennas

text: capable

text: of

text: frequency-controlled

text: x

text: x

text: beam

text: steering,

text: offering

text: high

text: directivity

text: and

text: simple

text: feeding

text: a

text: mechanism

text: With

text: the

text: advent

text: of

text: composite

text: right/left-handed

text: (CRLH)

text: leaky-wave

text: antennas

text: capable

text: of

text: full-space

text: scanning,

text: from

text: backward

text: to

text: forward

text: including

text: broadside

text: directions,

text: there

text: has

text: been

text: a

text: renewed

text: interest

text: in

text: such

text: structures

text: (b)

text: Full-scpae

text: scanning

text: was

text: made

text: possible

text: by

text: suppressing

text: the

text: c

text: b

text: open

text: stopband

text: and

text: equalizing

text: the

text: impedance

text: around

text: broad-

text: side

text: Following

text: these

text: advances,

text: periodic

text: leaky-wave

text: antennas

text: have

text: become

text: most

text: attractive

text: as

text: low-cost

text: alternative

text: to

text: phased

text: array

text: antennas

text: a

text: One

text: approach

text: to

text: achieve

text: a

text: full-space

text: beam

text: scanning

text: in

text: leaky-wave

text: structures,

text: is

text: to

text: use

text: the

text: phase-reversal

text: technique.

text: x

text: x

text: This

text: technique

text: involves

text: introducing

text: a

text: phase

text: shift

text: along

text: the

text: axis

text: of

text: the

text: structure,

text: so

text: that

text: λ

text: periodically

text: (c)

text: g

text: elements

text: get

text: excited

text: in

text: phase

text: and

text: therefore

text: radiate.

text: This

text: tech-

text: Fig.

text: Principle

text: of

text: phase-reversal

text: antenna

text: explained

text: in

text: terms

text: of

text: current

text: nique

text: is

text: common

text: in

text: slot-waveguide

text: antenna

text: arrays

text: where

text: the

text: distributions

text: in

text: two

text: adjacent

text: unit

text: cells.

text: a)

text: Differential

text: transmission

text: line

text: required

text: extra

text: phase-shift

text: is

text: induced

text: by

text: proper

text: location

text: without

text: phase

text: reversal.

text: b)

text: Phase

text: reversed

text: differential

text: lines

text: at

text: t

text: and,

text: c)

text: at

text: of

text: the

text: slots

text: along

text: the

text: waveguide

text: aperture

text: A

text: transmission-

text: t

text: where

text: T

text: is

text: the

text: time

text: period

text: of

text: an

text: assumed

text: time-harmonic

text: line

text: counterpart

text: of

text: this

text: structure

text: is

text: the

text: Sterba-curtain

text: antenna,

text: signal

text: of

text: frequency

text: which

text: utilizes

text: periodic

text: transmission-line

text: cross-overs,

text: placed

text: λ

text: apart,

text: to

text: produce

text: electric-dipole

text: radiation

text: from

text: the

text: cross-

text: While

text: the

text: radiation

text: mechanism

text: of

text: traditional

text: transmission-

text: g

text: overs

text: However,

text: these

text: waveguide

text: and

text: transmission-line

text: line

text: type

text: phase-reversal

text: antennas

text: is

text: well

text: understood

text: in

text: reso-

text: antennas

text: were

text: resonant

text: in

text: nature

text: and

text: therefore

text: featured

text: a

text: nant

text: conﬁgurations

text: and

text: extensively

text: discussed

text: in

text: the

text: literature

text: small

text: impedance-matching

text: bandwidth

text: product

text: This

text: travelling-wave

text: phase-reversal

text: antennas

text: have

text: not

text: been

text: phase

text: reversal

text: concept

text: was

text: later

text: extended

text: to

text: a

text: travelling-

text: exhaustively

text: explored.

text: Speciﬁcally,

text: they

text: have

text: been

text: described

text: wave

text: conﬁgurations,

text: thereby

text: both

text: improving

text: the

text: impedance-

text: in

text: terms

text: of

text: electric-dipole

text: radiation

text: only

text: In

text: this

text: work,

text: matching

text: bandwidth

text: performance

text: and

text: providing

text: full-space

text: the

text: radiation

text: mechanism

text: of

text: a

text: travelling-wave

text: type

text: phase

text: reversal

text: antenna

text: is

text: revisited

text: in

text: details

text: and

text: unveils

text: the

text: presence

text: S.

text: Gupta

text: and

text: L.

text: J.

text: Jiang

text: are

text: with

text: the

text: Electrical

text: and

text: Electronic

text: Engineering

text: of

text: magnetic-dipole

text: radiation

text: operating

text: in

text: conjunction

text: with

text: Department

text: of

text: University

text: of

text: Hong

text: Kong,

text: China

text: (SAR).

text: electric-dipole

text: radiation.

text: This

text: dual

text: radiation

text: mechanism

text: is

text: C.

text: Caloz

text: is

text: with

text: the

text: Department

text: of

text: Electrical

text: Engineering,

text: PolyGrames

text: Research

text: Center,

text: E´cole

text: Polytechnique

text: de

text: Montre´al,

text: Montre´al,

text: QC,

text: Canada.

text: speciﬁc

text: to

text: travelling-wave

text: type

text: phase-reversal

text: antennas

text: and

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 5 − − FIG. 2: Log log plot of the interval number N vs time t are FIG. 4: Log log plot of the interval number N vs yard-stick drawn for the kinetic DCS using three diﬀerent p values. The size δ using simulation data for three diﬀerent p values and the lines have slope equal to fractal dimension d = ln(1+p)/ ln 2, same initial condition. The lines have slopes d = p depicting ∼ f ∼ − f revealing that N(t) tdf as predicted by Eq. (18). that N(δ) δ df and hence this is in perfect agreement with our analytical result given by Eq. (25). FIG. 3: Plot of the pth moment M (t) of the remaining in- p the deﬁnition of the nth moment in the rate equation to tervals as a function time, where p is the fractal dimension of the stochastic DCS, showing that M (t) is a conserved quan- obtain (cid:104) (cid:105) p tity in the long-time limit. The three distinct parallel lines for the same probability p reveal that the numerical value of dM (t) − − (1 + p) the conserved quantity Mp(t), like kinetic DCS, is diﬀerent in n = 1 M (t). (23) each independent realization. dt (n + 1) n+1 Following the same procedure as for the kinetic DCS construction process are of equal size. In contrast, inter- problem, we obtain the asymptotic solution for the nth vals in the kinetic DCS are distributed not only randomly moment but they are also of diﬀerent size at any given stage. Yet, we ﬁnd that they share the same fractal dimension ow- ∼ − − ing to the same deterministic rule in the deﬁnition of the M (t) t(n p)z with z = 1, (24) n generator. Nature do not like determinism rather it likes to enjoy some form of freedom in the selection process. Freedom in the present context means the liberty to di- which implies that p is the special value of n for which vide an interval randomly. We therefore ask: What if we M (t) is a conserved quantity (see Fig (3)). p use a generator that divides an interval randomly into Note that the solution for M (t) given by Eq. (24) n two smaller intervals instead of dividing it into two equal clearly suggests that the dynamics of the stochastic DCS intervals? To this end we propose the following stochastic too governed by a conservation law namely the moment dyadic Cantor. We start the process with an initiator of of order n = p is a conserved quantity. The solution unit interval [0, 1] as before but unlike the previous cases for M (t) also suggests that the mean interval size δ(t) n the generator here divides an interval rando−mly into two decays following the same power-law and the same expo- pieces and removes one with probability (1 p). The al- nent as in Eq. (19). Using it as the yard-stick we ﬁnd gorithm of this model is exactly the same as the one for that the number N(δ) needed to cover the resulting set kinetic DCS except the fact that the generator always scales as divides an interval randomly into two parts instead of dividing into two equal parts. ∼ − We can make Eq. (7) describing the rules of the N(δ) δ df , (25) stochastic DCS problem if we only choose with d = p revealing that the index of the conserved f F(x, y) = 1, (21) quantity is again equal to the fractal dimension (see Fig. (4)). Note that the fractal dimension d = p is always f less than ln(1+p) regardless of the value of p. Hence we and at the same time replace the factor 2 of the gain can concludlen2that the fractal dimension of the stochas- term again by (1 + p). The master equation for the tic fractal is always less than that of its recursive or ki- stochastic DCS then is (cid:90) netic counterpart. Once again we ﬁnd that the moment of order equal to fractal dimension M is a conserved df ∞ quantity. It seems quite like a rule than otherwise as we ∂c(x, t) − = xc(x, t) + (1 + p) c(y, t)dy. (22) have found it to true also in the case of its opposition ∂t x phenomena namely in aggregation [15–20]. We shall now apply the Buckingham π-theorem to ob- The stochastic counterpart of the dyadic Cantor set tain a scaling solution for c(x, t) as it provides a deeper is much simpler to solve analytically than the stochas- insight into the problem [22]. Note that according to tic counterpart of the triadic Cantor set which was Eq. (22) the governed parameter c depends on three pa- ﬁrst proposed and solved analytically by Ben-Naim and rameters x, t and p of which only the former two are Krapivsky [20, 21]. However, we for the ﬁrst time give an dimensional. However, the knowledge about the decay exact algorithm of the model and focus primarily on ver- law for the mean interval size implies that one of the ifying the various analytical results by extensive numer- parameter, say x, can be expressed in terms of t since − ical simulation. Like before we once again incorporate we ﬁnd δ = t 1 must bear the dimension of the interval ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: Log

text: log

text: plot

text: of

text: the

text: interval

text: number

text: N

text: vs

text: time

text: t

text: are

text: FIG.

text: Log

text: log

text: plot

text: of

text: the

text: interval

text: number

text: N

text: vs

text: yard-stick

text: drawn

text: for

text: the

text: kinetic

text: DCS

text: using

text: three

text: diﬀerent

text: p

text: values.

text: The

text: size

text: δ

text: using

text: simulation

text: data

text: for

text: three

text: diﬀerent

text: p

text: values

text: and

text: the

text: lines

text: have

text: slope

text: equal

text: to

text: fractal

text: dimension

text: d

text: ln

text: same

text: initial

text: condition.

text: The

text: lines

text: have

text: slopes

text: d

text: p

text: depicting

text: f

text: f

text: revealing

text: that

text: N(t)

text: tdf

text: as

text: predicted

text: by

text: Eq.

text: that

text: N(δ)

text: δ

text: df

text: and

text: hence

text: this

text: is

text: in

text: perfect

text: agreement

text: with

text: our

text: analytical

text: result

text: given

text: by

text: Eq.

text: FIG.

text: Plot

text: of

text: the

text: pth

text: moment

text: M

text: (t)

text: of

text: the

text: remaining

text: in-

text: p

text: the

text: deﬁnition

text: of

text: the

text: nth

text: moment

text: in

text: the

text: rate

text: equation

text: to

text: tervals

text: as

text: a

text: function

text: time,

text: where

text: p

text: is

text: the

text: fractal

text: dimension

text: of

text: the

text: stochastic

text: DCS,

text: showing

text: that

text: M

text: (t)

text: is

text: a

text: conserved

text: quan-

text: obtain

text: p

text: tity

text: in

text: the

text: long-time

text: limit.

text: The

text: three

text: distinct

text: parallel

text: lines

text: for

text: the

text: same

text: probability

text: p

text: reveal

text: that

text: the

text: numerical

text: value

text: of

text: dM

text: (t)

text: p)

text: the

text: conserved

text: quantity

text: Mp(t),

text: like

text: kinetic

text: DCS,

text: is

text: diﬀerent

text: in

text: n

text: M

text: (t).

text: each

text: independent

text: realization.

text: dt

text: (n

text: Following

text: the

text: same

text: procedure

text: as

text: for

text: the

text: kinetic

text: DCS

text: construction

text: process

text: are

text: of

text: equal

text: size.

text: In

text: contrast,

text: inter-

text: problem,

text: we

text: obtain

text: the

text: asymptotic

text: solution

text: for

text: the

text: nth

text: vals

text: in

text: the

text: kinetic

text: DCS

text: are

text: distributed

text: not

text: only

text: randomly

text: moment

text: but

text: they

text: are

text: also

text: of

text: diﬀerent

text: size

text: at

text: any

text: given

text: stage.

text: Yet,

text: we

text: ﬁnd

text: that

text: they

text: share

text: the

text: same

text: fractal

text: dimension

text: ow-

text: ing

text: to

text: the

text: same

text: deterministic

text: rule

text: in

text: the

text: deﬁnition

text: of

text: the

text: M

text: (t)

text: t(n

text: p)z

text: with

text: z

text: n

text: generator.

text: Nature

text: do

text: not

text: like

text: determinism

text: rather

text: it

text: likes

text: to

text: enjoy

text: some

text: form

text: of

text: freedom

text: in

text: the

text: selection

text: process.

text: Freedom

text: in

text: the

text: present

text: context

text: means

text: the

text: liberty

text: to

text: di-

text: which

text: implies

text: that

text: p

text: is

text: the

text: special

text: value

text: of

text: n

text: for

text: which

text: vide

text: an

text: interval

text: randomly.

text: We

text: therefore

text: ask:

text: What

text: if

text: we

text: M

text: (t)

text: is

text: a

text: conserved

text: quantity

text: (see

text: Fig

text: p

text: use

text: a

text: generator

text: that

text: divides

text: an

text: interval

text: randomly

text: into

text: Note

text: that

text: the

text: solution

text: for

text: M

text: (t)

text: given

text: by

text: Eq.

text: n

text: two

text: smaller

text: intervals

text: instead

text: of

text: dividing

text: it

text: into

text: two

text: equal

text: clearly

text: suggests

text: that

text: the

text: dynamics

text: of

text: the

text: stochastic

text: DCS

text: intervals?

text: To

text: this

text: end

text: we

text: propose

text: the

text: following

text: stochastic

text: too

text: governed

text: by

text: a

text: conservation

text: law

text: namely

text: the

text: moment

text: dyadic

text: Cantor.

text: We

text: start

text: the

text: process

text: with

text: an

text: initiator

text: of

text: of

text: order

text: n

text: p

text: is

text: a

text: conserved

text: quantity.

text: The

text: solution

text: unit

text: interval

text: as

text: before

text: but

text: unlike

text: the

text: previous

text: cases

text: for

text: M

text: (t)

text: also

text: suggests

text: that

text: the

text: mean

text: interval

text: size

text: δ(t)

text: n

text: the

text: generator

text: here

text: divides

text: an

text: interval

text: rando−mly

text: into

text: two

text: decays

text: following

text: the

text: same

text: power-law

text: and

text: the

text: same

text: expo-

text: pieces

text: and

text: removes

text: one

text: with

text: probability

text: p).

text: The

text: al-

text: nent

text: as

text: in

text: Eq.

text: Using

text: it

text: as

text: the

text: yard-stick

text: we

text: ﬁnd

text: gorithm

text: of

text: this

text: model

text: is

text: exactly

text: the

text: same

text: as

text: the

text: one

text: for

text: that

text: the

text: number

text: N(δ)

text: needed

text: to

text: cover

text: the

text: resulting

text: set

text: kinetic

text: DCS

text: except

text: the

text: fact

text: that

text: the

text: generator

text: always

text: scales

text: as

text: divides

text: an

text: interval

text: randomly

text: into

text: two

text: parts

text: instead

text: of

text: dividing

text: into

text: two

text: equal

text: parts.

text: We

text: can

text: make

text: Eq.

text: describing

text: the

text: rules

text: of

text: the

text: N(δ)

text: δ

text: df

text: stochastic

text: DCS

text: problem

text: if

text: we

text: only

text: choose

text: with

text: d

text: p

text: revealing

text: that

text: the

text: index

text: of

text: the

text: conserved

text: f

text: F(x,

text: y)

text: quantity

text: is

text: again

text: equal

text: to

text: the

text: fractal

text: dimension

text: (see

text: Fig.

text: Note

text: that

text: the

text: fractal

text: dimension

text: d

text: p

text: is

text: always

text: f

text: less

text: than

text: regardless

text: of

text: the

text: value

text: of

text: p.

text: Hence

text: we

text: and

text: at

text: the

text: same

text: time

text: replace

text: the

text: factor

text: of

text: the

text: gain

text: can

text: the

text: fractal

text: dimension

text: of

text: the

text: stochas-

text: term

text: again

text: by

text: p).

text: The

text: master

text: equation

text: for

text: the

text: tic

text: fractal

text: is

text: always

text: less

text: than

text: that

text: of

text: its

text: recursive

text: or

text: ki-

text: stochastic

text: DCS

text: then

text: is

text: netic

text: counterpart.

text: Once

text: again

text: we

text: ﬁnd

text: that

text: the

text: moment

text: of

text: order

text: equal

text: to

text: fractal

text: dimension

text: M

text: is

text: a

text: conserved

text: df

text: quantity.

text: It

text: seems

text: quite

text: like

text: a

text: rule

text: than

text: otherwise

text: as

text: we

text: ∂c(x,

text: t)

text: xc(x,

text: t)

text: p)

text: c(y,

text: t)dy.

text: have

text: found

text: it

text: to

text: true

text: also

text: in

text: the

text: case

text: of

text: its

text: opposition

text: ∂t

text: x

text: phenomena

text: namely

text: in

text: aggregation

text: We

text: shall

text: now

text: apply

text: the

text: Buckingham

text: π-theorem

text: to

text: ob-

text: The

text: stochastic

text: counterpart

text: of

text: the

text: dyadic

text: Cantor

text: set

text: tain

text: a

text: scaling

text: solution

text: for

text: c(x,

text: t)

text: as

text: it

text: provides

text: a

text: deeper

text: is

text: much

text: simpler

text: to

text: solve

text: analytically

text: than

text: the

text: stochas-

text: insight

text: into

text: the

text: problem

text: Note

text: that

text: according

text: to

text: tic

text: counterpart

text: of

text: the

text: triadic

text: Cantor

text: set

text: which

text: was

text: Eq.

text: the

text: governed

text: parameter

text: c

text: depends

text: on

text: three

text: pa-

text: ﬁrst

text: proposed

text: and

text: solved

text: analytically

text: by

text: Ben-Naim

text: and

text: rameters

text: x,

text: t

text: and

text: p

text: of

text: which

text: only

text: the

text: former

text: two

text: are

text: Krapivsky

text: However,

text: we

text: for

text: the

text: ﬁrst

text: time

text: give

text: an

text: dimensional.

text: However,

text: the

text: knowledge

text: about

text: the

text: decay

text: exact

text: algorithm

text: of

text: the

text: model

text: and

text: focus

text: primarily

text: on

text: ver-

text: law

text: for

text: the

text: mean

text: interval

text: size

text: implies

text: that

text: one

text: of

text: the

text: ifying

text: the

text: various

text: analytical

text: results

text: by

text: extensive

text: numer-

text: parameter,

text: say

text: x,

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: t

text: since

text: ical

text: simulation.

text: Like

text: before

text: we

text: once

text: again

text: incorporate

text: we

text: ﬁnd

text: δ

text: t

text: must

text: bear

text: the

text: dimension

text: of

text: the

text: interval

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 value of the fractal dimension and found that it is always − FIG. 6: Log linear plots of the interval size distribution smaller than that of the DCS or the kinetic DCS. function c(x, t) of the kinetic DCS is drawn as a function of Besides giving exact analytical expressions for frac- x for three diﬀerent times. In inset we show that the three tal dimensions we have also given solutions for the in- distincts curves for three diﬀerent system sizes collapsed onto − terval size distribution function c(x, t) for both kinetic a single universal curve if we plot t (1+df)c(x, t) vs xt. It and stochastic DCS problem alongside numerical simu- implies that the system exhibits self-similarity with respect lations which fully support our solutions. In particu- to time. lar, we have shown that the distribution function c(x, t) evolves to a state where it exhibits dynamic scaling ∼ − a single universal curve which is essentially the scaling c(x, t) t1+pe xt. We used the idea of data-collapse to function φ(x). verify it numerically. For instance, we have drawn c(x, t) V. SELF-SIMILARITY IN KINETIC DCS vs x using data for a ﬁxed time and found a distinct set of curves for every diﬀerent time of the process. How- ever, these distinct curves collapsed onto a single univer- What about self-similarity in the kinetic dyadic Cantor sal curve when we expressed c(x, t) in unit of t1+p and x set? Does it also exhibit dynamic scaling? A closer look − ∼ − in unit of t 1. It implies that the system is similar to it- at the solution c(x, t) t1+pe xt for the stochastic DCS self at diﬀerent time and hence we say that such systems perhaps can guide us to write the solution of c(x, t) for exhibit self-similarity. It readily implies that the solution the kinetic DCS. To this end, we ﬁnd it instructive to is independent of initial condition. We have shown that express c(x, t) of stochastic DCS in terms of the mean the self-similar properties in all three cases are accompa- interval size δ(t) and the fractal dimension d f nied by conservation laws and interestingly the value of the fractal dimension d coincide with the index of the f c(x, t) ∼ δ−(1+df)φ(x/δ(t)), (36) conserved quantity Mdf . The numerical value of the con- served quantity M , however, is found diﬀerent at each df realization. We have then checked if the d th moment f ∼ − as we know δ t 1 and d = p for stochastic DCS. is also a conserved in the dyadic and traditional triadic f ∼ − That is, substituting d = ln(1 + p)/ ln 2 and δ t 1 Cantor set and found that it is obeyed in both the cases as f well. We think it is quite safe to argue that when systems for kinetic DCS we can immediately write the expected evolve and yet preserve self-similarity then the dynamics solution for it as of the system must be governed by conservation law. To explain why the quantity M always remains con- ∼ df c(x, t) t1+ln(l1n+2p) φ(xt). (37) stant we ﬁnd it useful to conside(cid:82)r a simple dimensional analysis. According to Eq. (35) the physical dimen- − sion of c(x, t) is [c] = L (1+df) s∞ince [s(t)] = L. To To verify it, we plot ﬁrst c(x, t) vs x for three diﬀerent know Why the quantity Mdf = 0 xdf c(x, t)dx a con- times (see Fig. (6)) and then plot the corresponding di- served quantity we ﬁnd it convenient to look into the mensionless quantities c(x, t)/t1+ln(l1n+2p) vs xt. We once physical dimension of its diﬀerential quantity dMdf = again ﬁnd a superb data-collapse which is shown in the xdf c(x, t)dx.−Using the physical dimension [x] = L and inset of the Fig (6) revealing that the fractal generated [c(x, t)] = L (1+df) in the expression for dMdf , we im- by kinetic DCS too exhibits a dynamic scaling vis-a-vis mediately ﬁnd that it bears no dimension and so does self-similarity. the quantity Mdf . It implies that the numerical value of VI. DISCUSSION AND SUMMARY Mdf remains the same despite the fact that the system evolves with time. On the other hand, the concentration c(x, t) is deﬁned as the number of particles per unit vol- In this section we attempt to summarize and put all ∼ ume of embedding space (V Ld where d = 1) per unit our ﬁndings into perspective. In this article we proposed − − mass (M) and hence [c] = L 1M 1 [25]. Now applying a dyadic Cantor set and then its kinetic and stochastic the principle of equivalence we obtain counterpart. In these models we incorporated probabil- ity, time and randomness in such a way that a logical ﬂow ∼ can easily be appreciated. Firstly, We solved the DCS M(L) Ldf , (38) problem by using probabilistic argument and found that it emerges as a fractal of dimension d = ln(1 + p)/ ln 2. f Then we solved the kinetic and stochasticc DCS using the which is often considered as one of the most commonly rate equation approach and found that the resulting sys- used benchmark for deﬁning fractal. An object whose tems emerge as fractal of dimensions d = ln(1 + p)/ ln 2 mass-length relation satisﬁes Eq. (38) with typically a f and d = p respectively. Our studies reveal that the non-integer exponent is said to be a fractal. f inclusion of kinetics alone does not alter the value of In summary, besides solving the model analytically, the fractal dimension. However, inclusion of randomness we also performed extensive numerical simulation which in the generator of the stochastic DCS does change the fully support all theoretical ﬁndings. Especially, the ##LTLine## ##LTLine##

text: value

text: of

text: the

text: fractal

text: dimension

text: and

text: found

text: that

text: it

text: is

text: always

text: FIG.

text: Log

text: linear

text: plots

text: of

text: the

text: interval

text: size

text: distribution

text: smaller

text: than

text: that

text: of

text: the

text: DCS

text: or

text: the

text: kinetic

text: DCS.

text: function

text: c(x,

text: t)

text: of

text: the

text: kinetic

text: DCS

text: is

text: drawn

text: as

text: a

text: function

text: of

text: Besides

text: giving

text: exact

text: analytical

text: expressions

text: for

text: frac-

text: x

text: for

text: three

text: diﬀerent

text: times.

text: In

text: inset

text: we

text: show

text: that

text: the

text: three

text: tal

text: dimensions

text: we

text: have

text: also

text: given

text: solutions

text: for

text: the

text: in-

text: distincts

text: curves

text: for

text: three

text: diﬀerent

text: system

text: sizes

text: collapsed

text: onto

text: terval

text: size

text: distribution

text: function

text: c(x,

text: t)

text: for

text: both

text: kinetic

text: a

text: single

text: universal

text: curve

text: if

text: we

text: plot

text: t

text: t)

text: vs

text: xt.

text: It

text: and

text: stochastic

text: DCS

text: problem

text: alongside

text: numerical

text: simu-

text: implies

text: that

text: the

text: system

text: exhibits

text: self-similarity

text: with

text: respect

text: lations

text: which

text: fully

text: support

text: our

text: solutions.

text: In

text: particu-

text: to

text: time.

text: lar,

text: we

text: have

text: shown

text: that

text: the

text: distribution

text: function

text: c(x,

text: t)

text: evolves

text: to

text: a

text: state

text: where

text: it

text: exhibits

text: dynamic

text: scaling

text: a

text: single

text: universal

text: curve

text: which

text: is

text: essentially

text: the

text: scaling

text: c(x,

text: t)

text: xt.

text: We

text: used

text: the

text: idea

text: of

text: data-collapse

text: to

text: function

text: φ(x).

text: verify

text: it

text: numerically.

text: For

text: instance,

text: we

text: have

text: drawn

text: c(x,

text: t)

text: V.

text: SELF-SIMILARITY

text: IN

text: KINETIC

text: DCS

text: vs

text: x

text: using

text: data

text: for

text: a

text: ﬁxed

text: time

text: and

text: found

text: a

text: distinct

text: set

text: of

text: curves

text: for

text: every

text: diﬀerent

text: time

text: of

text: the

text: process.

text: How-

text: ever,

text: these

text: distinct

text: curves

text: collapsed

text: onto

text: a

text: single

text: univer-

text: What

text: about

text: self-similarity

text: in

text: the

text: kinetic

text: dyadic

text: Cantor

text: sal

text: curve

text: when

text: we

text: expressed

text: c(x,

text: t)

text: in

text: unit

text: of

text: and

text: x

text: set?

text: Does

text: it

text: also

text: exhibit

text: dynamic

text: scaling?

text: A

text: closer

text: look

text: in

text: unit

text: of

text: t

text: It

text: implies

text: that

text: the

text: system

text: is

text: similar

text: to

text: it-

text: at

text: the

text: solution

text: c(x,

text: t)

text: xt

text: for

text: the

text: stochastic

text: DCS

text: self

text: at

text: diﬀerent

text: time

text: and

text: hence

text: we

text: say

text: that

text: such

text: systems

text: perhaps

text: can

text: guide

text: us

text: to

text: write

text: the

text: solution

text: of

text: c(x,

text: t)

text: for

text: exhibit

text: self-similarity.

text: It

text: readily

text: implies

text: that

text: the

text: solution

text: the

text: kinetic

text: DCS.

text: To

text: this

text: end,

text: we

text: ﬁnd

text: it

text: instructive

text: to

text: is

text: independent

text: of

text: initial

text: condition.

text: We

text: have

text: shown

text: that

text: express

text: c(x,

text: t)

text: of

text: stochastic

text: DCS

text: in

text: terms

text: of

text: the

text: mean

text: the

text: self-similar

text: properties

text: in

text: all

text: three

text: cases

text: are

text: accompa-

text: interval

text: size

text: δ(t)

text: and

text: the

text: fractal

text: dimension

text: d

text: f

text: nied

text: by

text: conservation

text: laws

text: and

text: interestingly

text: the

text: value

text: of

text: the

text: fractal

text: dimension

text: d

text: coincide

text: with

text: the

text: index

text: of

text: the

text: f

text: c(x,

text: t)

text: conserved

text: quantity

text: Mdf

text: The

text: numerical

text: value

text: of

text: the

text: con-

text: served

text: quantity

text: M

text: however,

text: is

text: found

text: diﬀerent

text: at

text: each

text: df

text: realization.

text: We

text: have

text: then

text: checked

text: if

text: the

text: d

text: th

text: moment

text: f

text: as

text: we

text: know

text: δ

text: t

text: and

text: d

text: p

text: for

text: stochastic

text: DCS.

text: is

text: also

text: a

text: conserved

text: in

text: the

text: dyadic

text: and

text: traditional

text: triadic

text: f

text: That

text: is,

text: substituting

text: d

text: p)/

text: ln

text: and

text: δ

text: t

text: Cantor

text: set

text: and

text: found

text: that

text: it

text: is

text: obeyed

text: in

text: both

text: the

text: cases

text: as

text: f

text: well.

text: We

text: think

text: it

text: is

text: quite

text: safe

text: to

text: argue

text: that

text: when

text: systems

text: for

text: kinetic

text: DCS

text: we

text: can

text: immediately

text: write

text: the

text: expected

text: evolve

text: and

text: yet

text: preserve

text: self-similarity

text: then

text: the

text: dynamics

text: solution

text: for

text: it

text: as

text: of

text: the

text: system

text: must

text: be

text: governed

text: by

text: conservation

text: law.

text: To

text: explain

text: why

text: the

text: quantity

text: M

text: always

text: remains

text: con-

text: df

text: c(x,

text: t)

text: φ(xt).

text: stant

text: we

text: ﬁnd

text: it

text: useful

text: to

text: a

text: simple

text: dimensional

text: analysis.

text: According

text: to

text: Eq.

text: the

text: physical

text: dimen-

text: sion

text: of

text: c(x,

text: t)

text: is

text: [c]

text: L

text: s∞ince

text: [s(t)]

text: L.

text: To

text: To

text: verify

text: it,

text: we

text: plot

text: ﬁrst

text: c(x,

text: t)

text: vs

text: x

text: for

text: three

text: diﬀerent

text: know

text: Why

text: the

text: quantity

text: Mdf

text: xdf

text: c(x,

text: t)dx

text: a

text: con-

text: times

text: (see

text: Fig.

text: and

text: then

text: plot

text: the

text: corresponding

text: di-

text: served

text: quantity

text: we

text: ﬁnd

text: it

text: convenient

text: to

text: look

text: into

text: the

text: mensionless

text: quantities

text: c(x,

text: vs

text: xt.

text: We

text: once

text: physical

text: dimension

text: of

text: its

text: diﬀerential

text: quantity

text: dMdf

text: again

text: ﬁnd

text: a

text: superb

text: data-collapse

text: which

text: is

text: shown

text: in

text: the

text: xdf

text: c(x,

text: t)dx.−Using

text: the

text: physical

text: dimension

text: [x]

text: L

text: and

text: inset

text: of

text: the

text: Fig

text: revealing

text: that

text: the

text: fractal

text: generated

text: [c(x,

text: t)]

text: L

text: in

text: the

text: expression

text: for

text: dMdf

text: we

text: im-

text: by

text: kinetic

text: DCS

text: too

text: exhibits

text: a

text: dynamic

text: scaling

text: vis-a-vis

text: mediately

text: ﬁnd

text: that

text: it

text: bears

text: no

text: dimension

text: and

text: so

text: does

text: self-similarity.

text: the

text: quantity

text: Mdf

text: It

text: implies

text: that

text: the

text: numerical

text: value

text: of

text: VI.

text: DISCUSSION

text: AND

text: SUMMARY

text: Mdf

text: remains

text: the

text: same

text: despite

text: the

text: fact

text: that

text: the

text: system

text: evolves

text: with

text: time.

text: On

text: the

text: other

text: hand,

text: the

text: concentration

text: c(x,

text: t)

text: is

text: deﬁned

text: as

text: the

text: number

text: of

text: particles

text: per

text: unit

text: vol-

text: In

text: this

text: section

text: we

text: attempt

text: to

text: summarize

text: and

text: put

text: all

text: ume

text: of

text: embedding

text: space

text: (V

text: Ld

text: where

text: d

text: per

text: unit

text: our

text: ﬁndings

text: into

text: perspective.

text: In

text: this

text: article

text: we

text: proposed

text: mass

text: (M)

text: and

text: hence

text: [c]

text: L

text: Now

text: applying

text: a

text: dyadic

text: Cantor

text: set

text: and

text: then

text: its

text: kinetic

text: and

text: stochastic

text: the

text: principle

text: of

text: equivalence

text: we

text: obtain

text: counterpart.

text: In

text: these

text: models

text: we

text: incorporated

text: probabil-

text: ity,

text: time

text: and

text: randomness

text: in

text: such

text: a

text: way

text: that

text: a

text: logical

text: ﬂow

text: can

text: easily

text: be

text: appreciated.

text: Firstly,

text: We

text: solved

text: the

text: DCS

text: M(L)

text: Ldf

text: problem

text: by

text: using

text: probabilistic

text: argument

text: and

text: found

text: that

text: it

text: emerges

text: as

text: a

text: fractal

text: of

text: dimension

text: d

text: p)/

text: ln

text: f

text: Then

text: we

text: solved

text: the

text: kinetic

text: and

text: stochasticc

text: DCS

text: using

text: the

text: which

text: is

text: often

text: considered

text: as

text: one

text: of

text: the

text: most

text: commonly

text: rate

text: equation

text: approach

text: and

text: found

text: that

text: the

text: resulting

text: sys-

text: used

text: benchmark

text: for

text: deﬁning

text: fractal.

text: An

text: object

text: whose

text: tems

text: emerge

text: as

text: fractal

text: of

text: dimensions

text: d

text: p)/

text: ln

text: mass-length

text: relation

text: satisﬁes

text: Eq.

text: with

text: typically

text: a

text: f

text: and

text: d

text: p

text: respectively.

text: Our

text: studies

text: reveal

text: that

text: the

text: non-integer

text: exponent

text: is

text: said

text: to

text: be

text: a

text: fractal.

text: f

text: inclusion

text: of

text: kinetics

text: alone

text: does

text: not

text: alter

text: the

text: value

text: of

text: In

text: summary,

text: besides

text: solving

text: the

text: model

text: analytically,

text: the

text: fractal

text: dimension.

text: However,

text: inclusion

text: of

text: randomness

text: we

text: also

text: performed

text: extensive

text: numerical

text: simulation

text: which

text: in

text: the

text: generator

text: of

text: the

text: stochastic

text: DCS

text: does

text: change

text: the

text: fully

text: support

text: all

text: theoretical

text: ﬁndings.

text: Especially,

text: the

text: ##LTLine##

text: ##LTLine##


text: (cid:88) − (cid:48) f(E ) f(E ) ∗ ∗ (cid:48) (cid:48) Π(r, r , ω) = 2e2 −m − −n ψ (r)ψ (r)ψ (r )ψ (r ), (5) E E ω iγ m n n m m n mn where f(E ) is the Fermi-function, ψ (r) is the energy eigen-function of electrons in n n the unperturbed system, and E is the eigen-energy. The unperturbed eigenstates can be n obtained using local dens(cid:88)ity functional theory. Substituting(cid:2) Eq.(5) in Eq.(1), (cid:3)we have − f(E ) f(E ) ∗ ρ(r, ω) = 2e2 −m − −n ψ (r)ψ (r) V ex (ω) + V in (ω) , (6) (cid:82) Em En ω iγ m n n,m nm mn ∗ where V X (ω) = drV X(r, ω)ψ (r)ψ (r).Combining Eq.(6) with Eqs.(4), we can obtain nm n m the self-consistent equation for V in(r, ω) (cid:90) (cid:88) (cid:2) (cid:3) − f(E ) f(E ) (cid:48) (cid:48) ∗ (cid:48) (cid:48) V in(r, ω) = 2e2 −m − −n dr K(r, r , ω)ψ (r )ψ (r ) V ex (ω) + V in (ω) , E E ω iγ m n n,m nm m n mn (7) (cid:48) | − (cid:48)| (cid:48) ∗ where K(r, r , ω) = 1/4πε r r + K (r, r , ω). Multiplying Eq. (7) by ψ (cid:48)(r)ψ (cid:48)(r) 0 xc m n and integrating(cid:88)over the space yields (cid:88) − [δ (cid:48) (cid:48) M (cid:48) (cid:48) (ω)] V in (ω) = M (cid:48) (cid:48) (ω)V ex (ω), (8) m n ,nm m n ,mn nm m n ,mn n,m mn mn (cid:90) (cid:90) with − f(E ) f(E ) (cid:48) ∗ (cid:48) ∗ (cid:48) (cid:48) M (cid:48) (cid:48) (ω) = 2e2 −m − −n dr dr ψ (cid:48)(r)ψ (cid:48)(r)K(r, r , ω)ψ (r )ψ (r ). m n ,mn E E ω iγ m n m n m n Now one can calculate the collective charge-oscillation (Eq.(6)) by solving Eq.(8). Setting V ex (ω) = 0, Eq.(8) becom(cid:88)es n,m − [δ (cid:48) (cid:48) M (cid:48) (cid:48) (ω)] V in (ω) = 0. (9) m n ,nm m n ,mn nm mn This is the plasmon eigen-equation we want, and with the equation all the plasmon ex- citations of a system may be found and are not dependent on the applied external ﬁelds. 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: f(E

text: f(E

text: Π(r,

text: r

text: ω)

text: −m

text: −n

text: ψ

text: (r)ψ

text: (r)ψ

text: (r

text: )ψ

text: (r

text: E

text: E

text: ω

text: iγ

text: m

text: n

text: n

text: m

text: m

text: n

text: mn

text: where

text: f(E

text: is

text: the

text: Fermi-function,

text: ψ

text: (r)

text: is

text: the

text: energy

text: eigen-function

text: of

text: electrons

text: in

text: n

text: n

text: the

text: unperturbed

text: system,

text: and

text: E

text: is

text: the

text: eigen-energy.

text: The

text: unperturbed

text: eigenstates

text: can

text: be

text: n

text: obtained

text: using

text: local

text: functional

text: theory.

text: in

text: have

text: f(E

text: f(E

text: ρ(r,

text: ω)

text: −m

text: −n

text: ψ

text: (r)ψ

text: (r)

text: V

text: ex

text: (ω)

text: V

text: in

text: (ω)

text: Em

text: En

text: ω

text: iγ

text: m

text: n

text: n,m

text: nm

text: mn

text: where

text: V

text: X

text: (ω)

text: drV

text: X(r,

text: ω)ψ

text: (r)ψ

text: (r).Combining

text: with

text: we

text: can

text: obtain

text: nm

text: n

text: m

text: the

text: self-consistent

text: equation

text: for

text: V

text: in(r,

text: ω)

text: f(E

text: f(E

text: V

text: in(r,

text: ω)

text: −m

text: −n

text: dr

text: K(r,

text: r

text: ω)ψ

text: (r

text: )ψ

text: (r

text: V

text: ex

text: (ω)

text: V

text: in

text: (ω)

text: E

text: E

text: ω

text: iγ

text: m

text: n

text: n,m

text: nm

text: m

text: n

text: mn

text: where

text: K(r,

text: r

text: ω)

text: r

text: r

text: K

text: (r,

text: r

text: ω).

text: Multiplying

text: Eq.

text: by

text: ψ

text: xc

text: m

text: n

text: and

text: the

text: space

text: yields

text: [δ

text: M

text: (ω)]

text: V

text: in

text: (ω)

text: M

text: (ω)V

text: ex

text: (ω),

text: m

text: n

text: ,nm

text: m

text: n

text: ,mn

text: nm

text: m

text: n

text: ,mn

text: n,m

text: mn

text: mn

text: with

text: f(E

text: f(E

text: M

text: (ω)

text: −m

text: −n

text: dr

text: dr

text: ψ

text: r

text: ω)ψ

text: (r

text: )ψ

text: (r

text: m

text: n

text: ,mn

text: E

text: E

text: ω

text: iγ

text: m

text: n

text: m

text: n

text: m

text: n

text: Now

text: one

text: can

text: calculate

text: the

text: collective

text: charge-oscillation

text: by

text: solving

text: Setting

text: V

text: ex

text: (ω)

text: n,m

text: [δ

text: M

text: (ω)]

text: V

text: in

text: (ω)

text: m

text: n

text: ,nm

text: m

text: n

text: ,mn

text: nm

text: mn

text: This

text: is

text: the

text: plasmon

text: eigen-equation

text: we

text: want,

text: and

text: with

text: the

text: equation

text: all

text: the

text: plasmon

text: ex-

text: citations

text: of

text: a

text: system

text: may

text: be

text: found

text: and

text: are

text: not

text: dependent

text: on

text: the

text: applied

text: external

text: ﬁelds.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [1] K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua and J. J. Baumberg, Nature (London) 491, 574 (2012). [2] J. Aizpurua, G. Hoﬀmann, S. P. Apell, and R. Berndt, Phys. Rev. Lett. 89, 156803 (2002). [3] P. Johansson, Phys. Rev. B 58, 10823 (1998). [4] D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. 90, 027402 (2003). [5] E. Prodan, C. Radloﬀ, N. J. Halas, and P. Nordlander, Science 302, 419 (2003). . [6] J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Kall, G. W. Bryant, and F. J. Garcia de Abajo, Phys. Rev. Lett. 90, 057401 (2003). [7] S. Gao and Z. Yuan, Phys. Rev. B72, 121406(R) (2005) [8] J. Yan and S. Gao, Phys. Rev. B78, 235413 (2008) [9] J. Yan, Z. Yuan, and S. Gao, Phys. Rev. Lett. 98, 216602 (2007) [10] N. Nayyar, V. Turkowski, and T. S. Rahman, Phys. Rev. Lett. 109, 157404 (2012) [11] A. E. DePrince, M. Pelton, J. R. Guest, and S. K. Gray, Phys. Rev. Lett.107, 196806 (2011) [12] R. F. Oulton,V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature (London) 461, 629 (2009). [13] K. Nakamura, K. Tanabe, and H. Atwater, Appl. Phys. Lett. 93, 121904 (2008). [14] K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, Nat. Mater. 3, 601 (2004). [15] N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005) [16] S. Nie and S. R. Emory, Science 275, 1102 (1997). [17] H. X. Xu, E. J. Bjerneld, M. K¨all, and L. B¨orjesson, Phys. Rev. Lett. 83, 4357 (1999) [18] A. T. Bell, Science 299, 1688 (2003). [19] E. Ozbay, Science 311, 189 (2006). [20] J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X. C. Yuan, F. Capasso, Science 340, 331 (2013) [21] L. R. Hirsch, R. J. Staﬀord, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, Proc. Natl. Acad. Sci. U.S.A. 100, 13549 (2003). [22] S. Kummel, K. Andrae, and P. G. Reinhard, Appl. Phys. B 73, 293 (2001) [23] N. Nilius, T. M. Wallis, and H. Ho, Science 297, 1853 (2002) 9 ##LTLine## ##LTLine## ##LTLine##

label: reference

text: K.

text: J.

text: Savage,

text: M.

text: M.

text: Hawkeye,

text: R.

text: Esteban,

text: A.

text: G.

text: Borisov,

text: J.

text: Aizpurua

text: and

text: J.

text: J.

text: Baumberg,

text: Nature

text: (London)

text: J.

text: Aizpurua,

text: G.

text: Hoﬀmann,

text: S.

text: P.

text: Apell,

text: and

text: R.

text: Berndt,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: Johansson,

text: Phys.

text: Rev.

text: B

text: D.

text: J.

text: Bergman

text: and

text: M.

text: I.

text: Stockman,

text: Phys.

text: Rev.

text: Lett.

text: E.

text: Prodan,

text: C.

text: Radloﬀ,

text: N.

text: J.

text: Halas,

text: and

text: P.

text: Nordlander,

text: Science

text: J.

text: Aizpurua,

text: P.

text: Hanarp,

text: D.

text: S.

text: Sutherland,

text: M.

text: Kall,

text: G.

text: W.

text: Bryant,

text: and

text: F.

text: J.

text: Garcia

text: de

text: Abajo,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Gao

text: and

text: Z.

text: Yuan,

text: Phys.

text: Rev.

text: J.

text: Yan

text: and

text: S.

text: Gao,

text: Phys.

text: Rev.

text: J.

text: Yan,

text: Z.

text: Yuan,

text: and

text: S.

text: Gao,

text: Phys.

text: Rev.

text: Lett.

text: N.

text: Nayyar,

text: V.

text: Turkowski,

text: and

text: T.

text: S.

text: Rahman,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: E.

text: DePrince,

text: M.

text: Pelton,

text: J.

text: R.

text: Guest,

text: and

text: S.

text: K.

text: Gray,

text: Phys.

text: Rev.

text: R.

text: F.

text: Oulton,V.

text: J.

text: Sorger,

text: T.

text: Zentgraf,

text: R.-M.

text: Ma,

text: C.

text: Gladden,

text: L.

text: Dai,

text: G.

text: Bartal,

text: and

text: X.

text: Zhang,

text: Nature

text: (London)

text: K.

text: Nakamura,

text: K.

text: Tanabe,

text: and

text: H.

text: Atwater,

text: Appl.

text: Phys.

text: Lett.

text: K.

text: Okamoto,

text: I.

text: Niki,

text: A.

text: Shvartser,

text: Y.

text: Narukawa,

text: T.

text: Mukai,

text: and

text: A.

text: Scherer,

text: Nat.

text: Mater.

text: N.

text: Fang,

text: H.

text: Lee,

text: C.

text: Sun,

text: and

text: X.

text: Zhang,

text: Science

text: S.

text: Nie

text: and

text: S.

text: R.

text: Emory,

text: Science

text: H.

text: X.

text: Xu,

text: E.

text: J.

text: Bjerneld,

text: M.

text: K¨all,

text: and

text: L.

text: B¨orjesson,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: T.

text: Bell,

text: Science

text: E.

text: Ozbay,

text: Science

text: J.

text: Lin,

text: J.

text: P.

text: B.

text: Mueller,

text: Q.

text: Wang,

text: G.

text: Yuan,

text: N.

text: Antoniou,

text: X.

text: C.

text: Yuan,

text: F.

text: Capasso,

text: Science

text: L.

text: R.

text: Hirsch,

text: R.

text: J.

text: Staﬀord,

text: J.

text: A.

text: Bankson,

text: S.

text: R.

text: Sershen,

text: B.

text: Rivera,

text: R.

text: E.

text: Price,

text: J.

text: D.

text: Hazle,

text: N.

text: J.

text: Halas,

text: and

text: J.

text: L.

text: West,

text: Proc.

text: Natl.

text: Acad.

text: Sci.

text: U.S.A.

text: S.

text: Kummel,

text: K.

text: Andrae,

text: and

text: P.

text: G.

text: Reinhard,

text: Appl.

text: Phys.

text: B

text: N.

text: Nilius,

text: T.

text: M.

text: Wallis,

text: and

text: H.

text: Ho,

text: Science

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 We consider the model (3) in a background of a uni- the vicinity of the transition temperature: B > B with − (cid:28) c form static magnetic ﬁeld parallel to the x axis. The B B B . One can show that in this case the ρ-meson 3 c c | | (cid:28) corresponding gauge potential is as follows: condensate is very small, ρ m , and the equations ρ of motion for ρ-meson ﬁ(cid:114)elds can be linearized [9]. The −B B A = y , A = x , A = A = 0 . (12) smallness parameter is: x 2 y 2 z t | | − g ρ (cid:39) B B We ignore quantum ﬂuctuations of the gauge and meson (cid:15) = s max c , B > B , (19) ﬁelds, thus treating the eﬀective model (3) at the classical mρ 2Bc c level. We always assume eB > 0 for simplicity. | | ≡ | | where ρ max ρ(z) is the maximal value of the The model (3) predicts that if the magnetic ﬁeld ex- max z inhomogeneous condensate of the ρ ﬁeld. ceeds the critical value (1) then the vacuum enters a new The classical equations of motion in the magnetic ﬁeld phase where positively and negatively charged ρ–meson background are discussed in details in Ref. [9]. The equa- ﬁelds condense [9]. In this phase the vacuum becomes tion for the charged ρ–meson condensate can be written a perfect electric conductor: the electric current can be as follows: carried by the ρ condensates along the lines of the mag- D netic ﬁeld without dissipation. The transport of electric ρ = 0 , (20) charge along the magnetic ﬁeld is described by a one- dimensional London equation, therefore we refer to the wOhere the correction to this equation is of the order of perfectly ideally conducting phase as “the superconduc- ((cid:15)4) with (cid:15) given in Eq. (19). Basically, the solutions tor phase” [9, 10]. of Eq. (20) reduce possible condensate solutions to the The electric superconductivity of the vacuum in strong space of the lowest Landau levels. ≡ magnetic ﬁeld is caused by a (p + ip)–wave condensation The neutral ρ–meson ﬁeld ρ(0) ρ(0) + iρ(0) can be ≡ (cid:104) (cid:105) x y of the charged ρ–meson ﬁeld ρ (x) u¯(x)γ d(x) with expressed via the charged ρ–meson condensate in the fol- − (cid:54) µ µ ρ = ρ = iρ = 0. Other vector components of the lowing form: x y vector condensate are zero, ρ = ρ = 0. In the mean– z t g ∂ | | ﬁeld approach and at the classical level of the eﬀective ρ(0) = i s − ρ 2 , (21) model (3), the superconducting ground state of the vac- 2 ∆ + m20 uum resembles the Abrikosov lattice state in an ordinary where ∆ = ∂∂¯ ≡ ∂2 + ∂2 is the two–dimensional Lapla- type–II superconductor [9, 10]. x y cian, The mean-ﬁeld solution for the ground state does not depend on the longitudinal z coordinate. Therefore it is 1 1 | | − (x⊥) = K (m x⊥ ) , (22) cpolenxvesnciaelnatrst:o∂co=mb∂in+e tih∂e,t∂r¯an=sv∂ers−e vi∂ec,toorrs iinntgoencoerma-l ∆ + m20 2π 0 O O O O x O −y O x y = + i , ¯ = i . As an exception to this is the two-dimensional Euclidean propagator of a scalar x y x y general convention we put particle with the mass of the neutral vector meson (10) and K is a modiﬁed Bessel function. − 0 ρ = ρ iρ , (13) Using the equations of motion for our model (3), the x y mean energy density (15) can be expressed via a nonlocal as the other combination simply vanishes in the LLL ap- function of the ρ–meson condensate ρ, proximation: ≡ (cid:104)E (cid:105) 1 (cid:104)E (cid:105) ρx + iρy 0 (LLL). (14) ⊥[ρ, ρ0, A] = 2B2 + ⊥[ρ] , (23) The static (potential) energy density co(cid:0)rresp(cid:1)onding to (cid:104)E⊥[ρ](cid:105) = e2 (cid:104)|ρ|2(cid:105)(cid:68)2 + 1(m2 − eB)(cid:104)|ρ(cid:69)|2(cid:105) the DSGS Lagrangian (3) is – assuming (14) – as follows: 8 2 ρ g2 | | 1 | | E⊥[ρ, ρ0, A] =(cid:0) 21|Dρ|2 + 12(cid:1)Fx2y + 12 ρ(x0y) 2 + 8s m2ρ ρ 2 −∆ + m20 ρ 2 , (24) +1m2 |ρ(0)|2 + |ρ|2 − e F ρ(0) , (15) where ρ ﬁeld is a(cid:104) mi(cid:105)nimum-energy solution of Eq. (20) 2 ρ g xy xy and the brackets . . . deno(cid:90)te the average over the trans- s verse plane: where D − (cid:104)O(cid:105) 1 O ρ = ∂ρ + ig ρ(0)ρ ieAρ, (16) = dxdy (x, y) , (25) s Area⊥ − i − (cid:0) (cid:1) Fxy = 2(∂¯A ∂A¯) = B , (17) and Area⊥ is the area of the transverse plane. g | | − i − The ground state solut(cid:88)ion of the ρ condensate in the ρ(x0y) = 2s ρ 2 2 ∂¯ρ(0) ∂ρ¯(0) . (18) B > Bc phase is given by the following expression [32], We consider the spontaneous condensation of the ρ me- ρ(x, y) = C h (x, y) , (26) n n son ﬁelds in the background of the magnetic ﬁeld (12) in ∈Z n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: consider

text: the

text: model

text: in

text: a

text: background

text: of

text: a

text: uni-

text: the

text: vicinity

text: of

text: the

text: transition

text: temperature:

text: B

text: B

text: with

text: c

text: form

text: static

text: magnetic

text: ﬁeld

text: parallel

text: to

text: the

text: x

text: axis.

text: The

text: B

text: B

text: B

text: One

text: can

text: show

text: that

text: in

text: this

text: case

text: the

text: ρ-meson

text: c

text: c

text: corresponding

text: gauge

text: potential

text: is

text: as

text: follows:

text: condensate

text: is

text: very

text: small,

text: ρ

text: m

text: and

text: the

text: equations

text: ρ

text: of

text: motion

text: for

text: ρ-meson

text: can

text: be

text: linearized

text: The

text: −B

text: B

text: A

text: y

text: A

text: x

text: A

text: A

text: smallness

text: parameter

text: is:

text: x

text: y

text: z

text: t

text: g

text: ρ

text: B

text: B

text: We

text: ignore

text: quantum

text: ﬂuctuations

text: of

text: the

text: gauge

text: and

text: meson

text: s

text: max

text: c

text: B

text: B

text: ﬁelds,

text: thus

text: treating

text: the

text: eﬀective

text: model

text: at

text: the

text: classical

text: mρ

text: c

text: level.

text: We

text: always

text: assume

text: eB

text: for

text: simplicity.

text: where

text: ρ

text: max

text: ρ(z)

text: is

text: the

text: maximal

text: value

text: of

text: the

text: The

text: model

text: predicts

text: that

text: if

text: the

text: magnetic

text: ﬁeld

text: ex-

text: max

text: z

text: inhomogeneous

text: condensate

text: of

text: the

text: ρ

text: ﬁeld.

text: ceeds

text: the

text: critical

text: value

text: then

text: the

text: vacuum

text: enters

text: a

text: new

text: The

text: classical

text: equations

text: of

text: motion

text: in

text: the

text: magnetic

text: ﬁeld

text: phase

text: where

text: positively

text: and

text: negatively

text: charged

text: ρ–meson

text: background

text: are

text: discussed

text: in

text: details

text: in

text: Ref.

text: The

text: equa-

text: ﬁelds

text: condense

text: In

text: this

text: phase

text: the

text: vacuum

text: becomes

text: tion

text: for

text: the

text: charged

text: ρ–meson

text: condensate

text: can

text: be

text: written

text: a

text: perfect

text: electric

text: conductor:

text: the

text: electric

text: current

text: can

text: be

text: as

text: follows:

text: carried

text: by

text: the

text: ρ

text: condensates

text: along

text: the

text: lines

text: of

text: the

text: mag-

text: D

text: netic

text: ﬁeld

text: without

text: dissipation.

text: The

text: transport

text: of

text: electric

text: ρ

text: charge

text: along

text: the

text: magnetic

text: ﬁeld

text: is

text: described

text: by

text: a

text: one-

text: dimensional

text: London

text: equation,

text: therefore

text: we

text: refer

text: to

text: the

text: wOhere

text: the

text: correction

text: to

text: this

text: equation

text: is

text: of

text: the

text: order

text: of

text: perfectly

text: ideally

text: conducting

text: phase

text: as

text: “the

text: superconduc-

text: with

text: given

text: in

text: Eq.

text: Basically,

text: the

text: solutions

text: tor

text: phase”

text: of

text: Eq.

text: reduce

text: possible

text: condensate

text: solutions

text: to

text: the

text: The

text: electric

text: superconductivity

text: of

text: the

text: vacuum

text: in

text: strong

text: space

text: of

text: the

text: lowest

text: Landau

text: levels.

text: magnetic

text: ﬁeld

text: is

text: caused

text: by

text: a

text: (p

text: ip)–wave

text: condensation

text: The

text: neutral

text: ρ–meson

text: ﬁeld

text: can

text: be

text: x

text: y

text: of

text: the

text: charged

text: ρ–meson

text: ﬁeld

text: ρ

text: (x)

text: u¯(x)γ

text: d(x)

text: with

text: expressed

text: via

text: the

text: charged

text: ρ–meson

text: condensate

text: in

text: the

text: fol-

text: µ

text: µ

text: ρ

text: ρ

text: iρ

text: Other

text: vector

text: components

text: of

text: the

text: lowing

text: form:

text: x

text: y

text: vector

text: condensate

text: are

text: zero,

text: ρ

text: ρ

text: In

text: the

text: mean–

text: z

text: t

text: g

text: ﬁeld

text: approach

text: and

text: at

text: the

text: classical

text: level

text: of

text: the

text: eﬀective

text: i

text: s

text: ρ

text: model

text: the

text: superconducting

text: ground

text: state

text: of

text: the

text: vac-

text: uum

text: resembles

text: the

text: Abrikosov

text: lattice

text: state

text: in

text: an

text: ordinary

text: where

text: ∂2

text: ∂2

text: is

text: the

text: two–dimensional

text: Lapla-

text: type–II

text: superconductor

text: x

text: y

text: cian,

text: The

text: mean-ﬁeld

text: solution

text: for

text: the

text: ground

text: state

text: does

text: not

text: depend

text: on

text: the

text: longitudinal

text: z

text: coordinate.

text: Therefore

text: it

text: is

text: (x⊥)

text: K

text: (m

text: x⊥

text: cpolenxvesnciaelnatrst:o∂co=mb∂in+e

text: tih∂e,t∂r¯an=sv∂ers−e

text: vi∂ec,toorrs

text: iinntgoencoerma-l

text: 2π

text: O

text: O

text: O

text: O

text: x

text: O

text: −y

text: O

text: x

text: y

text: i

text: i

text: As

text: an

text: exception

text: to

text: this

text: is

text: the

text: two-dimensional

text: Euclidean

text: propagator

text: of

text: a

text: scalar

text: x

text: y

text: x

text: y

text: general

text: convention

text: we

text: put

text: particle

text: with

text: the

text: mass

text: of

text: the

text: neutral

text: vector

text: meson

text: and

text: K

text: is

text: a

text: modiﬁed

text: Bessel

text: function.

text: ρ

text: ρ

text: iρ

text: Using

text: the

text: equations

text: of

text: motion

text: for

text: our

text: model

text: the

text: x

text: y

text: mean

text: energy

text: density

text: can

text: be

text: expressed

text: via

text: a

text: nonlocal

text: as

text: the

text: other

text: combination

text: simply

text: vanishes

text: in

text: the

text: LLL

text: ap-

text: function

text: of

text: the

text: ρ–meson

text: condensate

text: ρ,

text: proximation:

text: ρx

text: iρy

text: (LLL).

text: ⊥[ρ,

text: A]

text: ⊥[ρ]

text: The

text: static

text: (potential)

text: energy

text: density

text: to

text: the

text: DSGS

text: Lagrangian

text: is

text: assuming

text: as

text: follows:

text: ρ

text: E⊥[ρ,

text: A]

text: ρ

text: ρ

text: e

text: F

text: where

text: ρ

text: ﬁeld

text: is

text: solution

text: of

text: Eq.

text: ρ

text: g

text: xy

text: xy

text: and

text: the

text: brackets

text: the

text: average

text: over

text: the

text: trans-

text: s

text: verse

text: plane:

text: where

text: D

text: O

text: ρ

text: ∂ρ

text: ig

text: ieAρ,

text: dxdy

text: (x,

text: y)

text: s

text: Area⊥

text: i

text: Fxy

text: ∂A¯)

text: B

text: and

text: Area⊥

text: is

text: the

text: area

text: of

text: the

text: transverse

text: plane.

text: g

text: i

text: The

text: ground

text: state

text: of

text: the

text: ρ

text: condensate

text: in

text: the

text: ρ

text: ∂¯ρ(0)

text: ∂ρ¯(0)

text: B

text: Bc

text: phase

text: is

text: given

text: by

text: the

text: following

text: expression

text: We

text: consider

text: the

text: spontaneous

text: condensation

text: of

text: the

text: ρ

text: me-

text: ρ(x,

text: y)

text: C

text: h

text: (x,

text: y)

text: n

text: n

text: son

text: ﬁelds

text: in

text: the

text: background

text: of

text: the

text: magnetic

text: ﬁeld

text: in

text: ∈Z

text: n

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