About Superluminal Propagation of an Electromagnetic Wavepacket Inside a Rectangular Waveguide

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About Superluminal Propagation of an Electromagnetic Wavepacket Inside a Rectangular Waveguide

F. Cardone, R. Mignani, V. Olkhovsky

To cite this version:

F. Cardone, R. Mignani, V. Olkhovsky. About Superluminal Propagation of an Electromagnetic

Wavepacket Inside a Rectangular Waveguide. Journal de Physique I, EDP Sciences, 1997, 7 (10),

pp.1211-1219. �10.1051/jp1:1997118�. �jpa-00247393�

About Superluminal Propagation of

_an

Electromagnetic Wavepacket Inside

_a

Rectangular Waveguide

F.

Cardone (~),

R.

Mignani

(~>~>*) and

V-S- Olkhovsky (~)

(~ Universith

Gregoriana

and

GNFM-CNR,

Piazza delta Pilotta

4,

00187

Roma, Italy

(~)

Dipartimento

di Fisica

"E.Amaldi",

Universith

degli

Studi "Roma Tre",

via delta Vasca Navale 84, 00146

Roma, Italy

(~) INFN, ^{Sezione di Roma} I,

c/o

Dipartimento di Fisica, I Universit£

degli

Studi di Roma

"La Sapienza", P.le A.More 2, 00185 Roma,

Italy

(~) Institute for Nuclear Research, Ukrainian

Academy

of

Sciences,

Prospect Nauki 47, 252028 Kiev, Ukraine

(Received

22 November

1996,

revised 21

May

1997,

accepted

9 June

1997)

PACS.03.40.Kf Waves and _wave propagation; general mathematical aspects PACS.03.50.De Maxwell

theory: general

mathematical aspects

PACSAI.20.Jb

Electromagnetic

_wave

propagation:

radiowave

propagation

Abstract. We discuss the

propagation

of _an

electromagnetic wavepacket

inside _a rectangular

waveguide,

of the type

employed

in recent experiments _on

superluminal tunneling

of

electromag-

netic

signals. By exploiting

the

analogy

between particle and photon

tunneling,

_we consider both evanescent and

growing

_waves inside the narrowed part of the

waveguide.

The Fourier expansion of such _waves shows that the barrier behaves in _a nonlocal way. Such _a

nonlocality

is accounted

for in _an effective _way

by

_means of _a deformation of the spacetime ^{inside the}

waveguide.

As _a consequence, the wavepacket propagates at

superluminal speed according

to _an effective metric tensor, built _up in

analogy

with the Cauchy stress tensor in _a deformable medium.

1. Introduction

In the last _years, there has been _a renewed interest _on

superluminal

_processes, due to some new

experimental

evidences in different sectors of

physics.

Those

include,

_e-g-, the

apparent superluminal expansions

of

galactic objects iii

and the evidence for

superluminal

motions in

electrical and acoustical

engineering [2j. However, perhaps

the most

interesting experimental findings

are those

concerning

the

superluminal tunneling

of evanescent _waves and

photons [3-7],

first observed at

Cologne

_[3] and

Berkeley _[5j,

and then confirmed

by

_a Florence [6j and _a Vienna _{[7] group.}

From the theoretical

point

^of

view,

evanescent _{waves were}

predicted

to be

superluminal

_[8j

on the basis of the

analogy

between

tunneling photons

and

tunneling particles

_[9j

(which,

_as

is well

known,

_can move with

superluminal speed

inside the barrier the so-called Hartmann effect

[10j ).

Some aspects ^{of the}

superluminal propagation

of

electromagnetic wavepackets

_were

discussed in references

[8-lsj.

(*)

Author for correspondence

(e-mail: mignani©romal.infn.it

_or

mignani©amaldi.fis.uniroma3.it)

@

Les

iditions

_de

_Physique

₁₉₉₇

1212 JOURNAL DE

PHYSIQUE

I N°10

2 2

Fig.

I. Rectangular waveguide with variable section. L

=

length

of the smaller

waveguide;

b = width of the smaller

waveguide;

the numbers I and 2 refer,

respectively,

to the regions inside the

larger

and the smaller waveguide.

In the

present

paper, we want to discuss in detail the

superluminal propagation

^of an elec-

tromagnetic wavepacket

inside _a

rectangular waveguide.

The main tools _we shall

exploit

to this aim _are: the

analogy

between

particles

and _waves [9,

lo,16]

and the formalism of deformed

Minkowski _space

[17-19].

The paper is

organized

_as follows. In Section 2 _we introduce the basic notions

concerning

the

propagation

of the

electromagnetic wavepacket

inside the

waveguide, by exploiting

the

analogy

between

particle

and

photon tunneling.

In Section 3 _we

Fourier-expand

the

wavepacket

inside

the _narrow

part

^of the

waveguide,

calculate the

partial

fluxes and time

delays,

and show that

the barrier behaves in _a nonlocal way. Such _a

nonlocality

is

described,

in Section 4, in terms of _a deformation of the Minkowski

space-time,

which allows _us to account for the

superluminal

propagation

of each Fourier

component

^{of the} wave

packet.

Section 5 concludes the paper.

2. Helmholtz

Equation

for _an

Electromagnetic Wavepacket

Let _us consider _a hollow

rectangular waveguide

with variable section

(like

that used in the

Cologne experiment [3]),

^{where the} narrow

part

^has

length L, height

b and thickness _a

(see Fig. I).

Inside the

waveguide,

_any of the vector

quantities

f

describing

the

electromagnetic

field

if

=

A, E,

_or

H,

where A is the vector

potential

with the

subsidiary

_gauge condition div A

= 0, E

=

ii /c)9A fat

is the electric field and H

= rot A is the

magnetic field)

is ruled

by

the Helmholtz

equation

nf

=

i7~f ~~

= 0.

(2.I)

The

general

solution of

equation (2.I)

is

given by

_a

wavepacket

of the kind

f(r,t)

=

~

~x(k)ak(r)e~~~~~ (2.2)

k>o)

where,

_as

usual, ko

_#

w/c

=

e/hc,

k

=

(k(

=

ko,

and

2

xlk)

=

ljx~lk)e~ik) 12.3a)

e~ e~ =

i~ e~jk)

k

=

o, i, j

₌

1,

^2.

(2.3b)

The vectors ej are the

(linearly independent) polarization vectors,

and

x~(k)

is the

amplitude

for the

photon

to have momentum k and

polarization I,

_so that

jX~(k)j~d3k

is

proportional

to

the

probability

that the

photon

has _a momentum between k and k ₊ dk in the

polarization

state e~.

Moreover,

_we have

(assuming

the z-axis

along

the

waveguide)

ak(r)

=

~~~ ~ + ~Re ~~~~+~~~~+~~YY

region

1

(°~

^~~ +

fle~~~)e~~~~+~kYY region

2

(~.~)

i._{e. a} linear

superposition

of

plane

_waves in

region

I and _a linear

superposition

of _an evanescent and _an

increasing

_wave in

region

2. On account of the

boundary

conditions and for transverse

electric

(TE)

_waves the

following

relations hold:

k~

=

), _kg

=

); lm,nintegersl _12.51

iii~

₊

iii~ iii~ ₁₂₆₁

x = 27r

Ill

^~

li)~

=

~= _Ill

l~il~)~

^~2

^12.71

where

lc

is the cutoff

wavelength.

Let _us

explicitly

notice that the need of

taking

into account both evanescent and

growing

waves in

region

2 is demanded

by

the

analogy

between

photon

and

particle tunneling [16].

By

the _same

analogy,

_one

gets

^{the last}

expression

_{of x in}

equation (2.7);

_moreover, it _can be further

exploited

in the calculation of the

stationary

flux.

Indeed,

_we

have,

for

particles

^of

mass p and

wavefunctionlfi(z):

~~

~~~~~ ^'

~~'~~

On the other

side,

the

z-component

of the

Poynting

vector is

given by

Sz

=

~Re[E*

x

H]z

=

iRe[E](~~~ ^~~~ _Ej(~~~ ^~~~ _{ii. (2.9)}

47r 41r @z fix

8y

@z

For monochromatic TE waves, ^we have

~

=

-Eo[

CDs

k~z

sin

kyve~~~tizi

Ey

₌

Eo

sin

k~z

_cos

kg ye~"~ifi (z) (~.1°)

Ez

= 0

~~'~~~~

lb(Z)

_~

~~~xll~~+~~~~ ~

_{~~ ~~~}

Therefore,

the vector

potential

is

A~

= a~

ix, y)ifi(z)e~~"~

Au

= ay

lx, vl~filzle~~"~ 12.121

Az

= 0

with

)~~~~~

=

j~~~~~~~~~~ _~~'~~~

1214 JOURNAL DE

PHYSIQUE

I N°10

Replacing equations (2.12, 2.13)

in

(2.9),

_we

get

the

following expression

of the

Poynting

vector

component:

~~

_~~~~~~^~ _{~ ~~~} ^~ _~~~~~

~~ ~

~'

~~'~~~

Therefore,

the flux

Sz

for

photons

is obtained from the flux for

particles by simply replacing

in

equation 12.81 1-ih/2vl by lla~l~

+

layl~ll-iUJ/47rl.

3. Fourier

Analysis

of the

Wavepacket Propagation

We want _now to discuss in detail the

propagation

of the

electromagnetic wavepacket

inside the

narrow

part

of the

waveguide (region 2).

Let _us first _consider the evanescent wave in

region 2,

given by equation (2.4b),

and

expand

it in _a Fourier series:

«

e~X~

=

~j one~"~~~/~ (3.1)

n=-m

where

(for XL

»

Ii

cm =

~~~~(~ (3.2)

X +

in~

We _can calculate the

stationary

flux of the monochromatic

wavepacket by exploiting

the anal- ogy with

particle tunneling

_as shown in the

previous

section. So the flux for the _evanescent

wave

(3. Ii

is

given by (2.8).

After

lengthy

but

straightforward calculations,

we

get

~~

~

^~

~

_~ ^{~~~~ ~}

^{~~~ ~}

^~~~

~~i~l~l'lli12~lm21i11 ^{~~~'~~ ~i + zii}

~°~~~~ ~~~~

~~ ~~

" + +

~ ~ ~ ~

m,n m=n m>n m<n

#- ~ + =0.

~ ~ ~ ~

m>n m<n m>n m<n

~~~~~~°~~

7rh

(1

~

f

n

o

(3.4)

~ "

J II

~~_~

x~

₊

n~lf)~

as

expected. So,

_{we can}

analyze only

the

partial

fluxes

jn, given by

Jn

=

~l11)~x~

+

ij+j~ ^13.5)

where _{n >} 0

corresponds

to

incoming

waves, and _{n <} 0 to

returning

_waves.

For

quasi

monochromatic _wave

packets,

^{the n-th} term of the _sum in

(3. Ii

_must be

replaced by

~

~l~n2nz/L)-j«t/h) j~ ~j

Then,

it is easy ^to

get, by

a standard

procedure (in

the

stationary phase approximation) _[9],

the

partial

^time

delays:

~~" ~~~~

^~~

~~~~~~~

~

i +

yin _j2 ~ ) _(3.7j

which,

_as is well

known,

do

represent

^the

delays

in the arrival of the _wave maxima.

Since

~ ^~'

~~'~~

~~ ~~~ ~~~~~Y

A7n

= _h

27rnkz

~CXIX~

+

~j"j2j' ^j~

^~~

We have

therefore

_that

(like

in the

particle case) ii

_n < 0

corresponds

to a

delay

of the _wave;

iii

n > 0

corresponds

to _an advance of the _wave.

For the

growing

_wave e+X~ with Fourier series

m

e+X~

=

~j due~"f~, _(3.10)

n=-«

we

get (since

_{x ~}

-xi analogous

but

interchanged

results.

On the basis of the above

results,

_{we can} therefore conclude that:

a)

The evanescent _wave

(3. Ii

contains two kinds of _waves:

incoming (moving

toward the

positive

direction of the

z-axis)

and returned

ii.

_e.

reflected by

the barrier

region

2 _{as a}

whole);

b)

In the

growing

_wave

(3.10),

there _are still two kinds of _waves, but of

origin different

from the evanescent _case,

namely:

_waves

reflected by

the "second barrier well"

(junction

2 _~

Ii

and

stcondly

returned

by

^the barrier

region

2 _{as a} whole

ii.

e. still

moving

ⁱⁿ the

positive

z-axis

direction,

_as the

initially incoming ones).

Thus,

the linear combination

(2.lib)

of the _evanescent and the

growing

_wave contains all the above

four

kinds

of

current _waves.

They give

^rise to nonzero

fluxes j+, j- (where _{j+, j-}

_are

the

positive

and the

negative component

^{of the total}

flux-according

to reference [9j associated with motion

along

the

positive

and the

negative z-direction, respectively ).

By

the way, in all the _sums the _{term n}

= 0

corresponds

to zero

flux.

This may be inter-

preted

_as

representing

the

time-integral probability of particle (photon) dwelling,

due to the

accumulating (to

and

from)

current waves.

4.

Nonlocality

of

Propagation

and Deformed Minkowski

Space

Let _us notice that the behaviour of the barrier

(region 2)

is nonlocal. This is due to the fact that

(both

in the _case of the evanescent and the

growing wave)

the _{waves are}

reflected by

the

barrier _{as a} whole

(see points a), b)

of the

previous section).

Such nonlocal effects in the

propagation

^{of the}

wavepacket

allow _us to

clarify

_a basic

point

of _our

approach. Indeed,

what is the

physical meaning

of the real momenta which _appear in the Fourier transforms of the evanescent and the

growing

wave?

This

point

can be understood

by noting

that nonlocal effects inside _a narrowed

waveguide

_can

be taken into account in _an effective _way

by

_means of _a deformed Minkowski

spacetime iii,18j.

Namely, according

to reference

iii],

_an evanescent mode in the usual Minkowski _space is described _{as a} non-evanescent one, with a real

wavevector, propagating

at _a

superluminal speed

in _a deformed Minkowski _space endowed with metric

q =

diag(b(, -b(. -b(, -b() _(4.1)

where the metric

parameters b(

are functions of the energy of the

single photon:

b(

₌

b((E),

_{~ =}

0,1, 2,

3.

(4.2)

1216 JOURNAL DE

PHYSIQUE

I N°10

In _our case, E

corresponds

to the _energy of the Fourier

component (see below).

Wave

propagation

in such _a

spacetime

^{is described}

by

the deformed Helmholtz

equation [17,18j

Ananalysis

of the energy

of

the

metric

parameters ^performed by

sing

the _data of _the

reported in Ref.

^{hows that}

the

^etric q ^educes to the^sual

Minkowskian

one, g = diag(1, -1, -1,

-

Ii, ^{for energies E >}

Eo

~

4,

case, the metric

is

_fully

Minkowskian

in region

I,

and_becomes

deformed

in _region 2

E

< Eo)

_Iii].

In

any _Fourier mponent

of the

_avepacket

satisfies

the deformed

elmholtz equation

(4.3)

_{in region} 2.

Without

loss of generality (since all quantities of

nterest

to us depend on the atio

b(/b(), we

can

_assume ^that the deformed ^etric is _sochronous with

the

usual one,

_i-e-

b(

11)"~l~

₌

(~)) (~) ^fj

14.41

~

b3

~

where _uc is the cutoff

frequency.

Since

ji("1

=

27rn/L,

_we

_get

the

explicit expression

^of the parameter

b)~~.

~(")

~

"

=

L"16'~

j4 _~)

~

u)

+

(nc/L)2

c2 ₊

(Luc/n)2'

The

speed

_un of the n-th Fourier

component

^therefore reads

Since

~

> i

14.71

~~/n

for

v < u~

(4.8)

(and

_n >

0),

_we

always

have

un ^> c

(4.9)

I.e. all the

propagating

_waves ⁱⁿ the Fourier

expansions (3.1), (3.10)

_are

superluminal.

Let

us recall that the

"superluminality

condition"

(4.8) iii]

is in fact satisfied

by

the

parameter

values of the

Cologne experiment (with (v~/v)~

=

l.2).

Notice that each Fourier

component propagates

ⁱⁿ a

different deformed

Minkowski _space- time. This is

clearly

related to the _energy

(and momentum) dependence

of the

parameters

^of the deformed metric

(4.I).

If

q$~

is the deformed metric "seen"

by

the n-th Fourier

component

of the evanescent wave, we can build _{up an}

effective

metric tensor

jjp~

^{for the} evanescent wave

as follows:

~j _lcnl~n)]~

~""~~"~

~~j lcnl~

^~~

~°~~

n

where the

cm's

_are the coefficients of the Fourier

expansion (3. Ii.

The

analogous

definition for the

growing

_wave is

~j'~n(~il~i~

~""~~~~

^~

j~

_j2

~~'~~~~

~

n n

where the

d[s

_are the coefficients of the

expansion (3.10).

Notice

that,

in

general,

the n-th Fourier

component

^of the evanescent and the

growing

wave will _{see a}

different

metric deforma- tion

ii-

e.

q$~ # q)0~)

due _to the connection among the wavevector and the metric

parameters (see ^Eq. (4.4)),

and the

dependence

of the latter _{ones on} the energy

(Eq. (4.2)).

The total effect of metric deformation is therefore

represented by

_a linear combination of the tensors

(4.10)

for both the evanescent and the

growing

_wave:

~j _lcnl~§# ~j _ldnl~§)l~

win

_~

1°l~ivmlcnl

+

lfll~ivmldnl

_~

1°l~ "~

~

+

lfll~ "~

~

14.ill '~"' '~"'

where _a,

fl

_are the coefficients in

equation (2.lib).

Clearly,

in

region

1

(where

all Fourier _waves do

propagate

in _a Minkowskian

spacetime),

_one

recovers, from definitions

(4.10, 4.11),

the usual metric _g.

In

fact,

in

region

1we have E _>

Eo,

so that

b((E)

=

1,

_~

=

0,1, 2,

3

(see

Refs.

iii,18]),

and therefore

q$~

=

g~~vn. So,

^all definitions

(4.10, 4.ll)

reduce to the Minkowskian metric.

Notice that the tensors

(4.10)

_are

analogous

to the

Cauchy

stress tensor.

Indeed,

let _us

consider,

in

orthogonal

Cartesian

coordinates,

an infinitesimal tetrahedron with

edges parallel

to the coordinate _axes and the

oblique

face S

opposite

to the vertex

0, origin

of the Cartesian frame.

If the tetrahedron is _a part of _a continuous

body,

the stress vector _across S in the

point

0 is

given by _[20]

~§ ~~

~

(~ba)o

#

f

~

(4.12)

ja~j

~

where _a is _a vector normal to S and (q§~)o

Ii

=

1, 2, 3)

is the stress vector on the face of the tetrahedron

orthogonal

to the I-th axis. The nine

components

^{of the three} vectors (q§~)o ^do

just

constitute the

rank-two, symmetric Cauchy

tensor.

The _tensor

jj

can be therefore

regarded

_as the average tensor

representing

the

space-time

deformation in the

region

2 of the

waveguide (corresponding

to the energy E _<

Eo

~ 4.5

~eV)

globally

"seen"

by

the

electromagnetic

Fourier

components

for the

wavepacket (2.I16 _). So,

_we

can name it average ^tensor

of

the

electromagnetic space-time deformation,

_{ije_m_.}

It is worth

stressing

that _our

approach

to

electromagnetic faster-than-light propagation

in

waveguides

is

similar,

in _some respects, to that where

superluminal propagation (e.g.

^of

^light

between

parallel mirrors)

is connected to vacuum effects

[21].

^{In such} a case, the influence of the

(structured)

_vacuum is described in _an effective _way in terms of _a refractive index

(as pioneered

by Sommerfeld). Something analogous

does

happen

in General

Relativity,

too: the deflection of

light

_rays in _a

gravitational

field _can be considered _{as a}

propagation

in _an Euclidean _space, filled with _a medium endowed with _an effective refractive index

[22, 23].

In _some cases, such

1218 JOURNAL DE

PHYSIQUE

I N°10

a

propagation,

due to the influence of the

gravitational

_vacuum, turns out to be

superluminal (the

refractive index is less than

one) _[24].

Notice that _our

approach

_can be therefore

regarded

as dual to the

general

relativistic _one, in which the

spacetime

curvature for

electromagnetic signals

is

replaced by

a refractive index.

Indeed,

in _our

formalism,

the _{vacuum or} nonlocal

effects

which

affect propagation

in the

wavegmde

_are

directly

described in terms

of

_a

spacetime deformation (and

the role of the refractive index is

played by

^the deformation

tensor).

Moreover,

it is

easily

_seen that definitions

(4.10, 4.ll)

_can also be

applied

to

wavepackets

ruled

by

interactions different from the _{e-m- one}

[17,19]. ^Namely,

we can state in full

generality

that the Minkowski space is

always subjected

to _a

stress,

^{whenever crossed}

by

_a

wavepacket.

Such _a stress

produces

_a deformation of the

spacetime,

which _may be described

by

the tensor fj _{= g}

(ineffectual deformation)

_or

by

a tensor

jj #

_g

(effectual deformation).

^The two _cases

j

_{= g,}

j #

_{g are}

obviously

determined

by

the interaction

ruling

the

wavepacket propagation

and

by

the energy of the

wavepacket components (see Eq. (4.2)).

The

superluminal propagation,

which is _an

unescapable

consequence of the _tensor

(4.10)

in

case of effectual

deformation,

does _no

longer imply

_a violation of

causality

in the deformed Minkowski _space

(~).

This is in fact related to the invariance of the tensor

(4.10)

under the

generalized

Lorentz transformations valid in the deformed Minkowski _space

(see

^Refs.

[16, Ii,19]).

5. Conclusions

In this paper, we have discussed the

propagation

of _an

electromagnetic wavepacket

inside _a hollow

waveguide

with variable section. On the basis of the

analogy

between

particle tunneling

and

photon tunneling,

_we have shown that _a

complete analysis

of the process

requires

consid-

ering

both _an evanescent and _a

growing

_wave inside the

waveguide.

We have

Fourier-analyzed

such _waves, and stressed that both of them contain returned waves, which _are due to the action of the barrier

(narrowed part

of the

waveguide)

_{as a} whole. Such _a behaviour of the barrier is

therefore nonlocal. As _a consequence, we can

reinterpretate

^the

propagation

of each Fourier

component

inside the barrier _{as a wave}

propagation

in _a deformed Minkowski

spacetime.

This

implies

_a real wavevector for each Fourier wave, and

superluminal propagation

of each Fourier

component.

So _our

approach

^leads in _a

straightforward

_way to

explaining

^the

superluminal tunneling

of the

wavepacket

_as its

propagation

in _a

region

^{with deformed} metric. Such _{a space-} time deformation _can be

described,

in _an effective _way,

by

_{an average}

symmetric

tensor

jp~,

built _up from the deformed metrics "seen"

by

each Fourier

component

^{of the}

wavepacket,

in

analogy

with the well-known definition of the

Cauchy

stress tensor for _a deformable medium.

Acknowledgments

We _are very

grateful

to W. Heitmann and G. Nimtz for

precious

comments and

bibliographical

advice, and to the referees for their

criticism,

that allowed _us to

considerably improve

the paper. One of _us

(V.S.O.)

thanks the

Department

of

Physics

"E. Amaldi" of Rome

University

"Roma Tre" for financial

support,

and the

hospitality

extended to him.

(~) ^{This is related to the fact}

that,

inside _a deformed Minkowski _space, the maximal causal

speed is,

in

general,

greater than _c,

depending

_on the metric parameters. See references

[18,19j,

and references therein.

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