Traversal Time as a Function of the Size of the Wavepacket

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Traversal Time as a Function of the Size of the Wavepacket

J. Ruiz, M. Ortuño, E. Cuevas, V. Gasparian

To cite this version:

J. Ruiz, M. Ortuño, E. Cuevas, V. Gasparian. Traversal Time as a Function of the Size of the Wavepacket. Journal de Physique I, EDP Sciences, 1997, 7 (5), pp.653-661. �10.1051/jp1:1997182�.

�jpa-00247351�

Traversal Time

_{as a}

Fuuction of the Size of the Wavepacket

J. Ruiz

(~,*),

^M.

^0rtuilo (~),

E.

Cuevas (~)

^{and V.}

Gasparian _(~)

(~)

Departamento

de

Fisica,

Universidad de lvfurcia, Murcia,

Spain

(~)

Department

of

Physics,

University ^of

Yerevan,

Yerevan, ^Armenia

(Received

29

July

1996, received in final form 22

January 1997,

accepted 28

Jauuary 1997)

PACS.41.20.Jb

Electromagnetic

wave

propagation;

radiowave

propagation PACS.42.25.-p

Wave

optics

Abstract. We calculate

numerically

the transmission coefficient and the traversal time for finite Gaussian

wavepackets

_as a function of their widths. We consider

electromagnetic

_waves

crossing

_a slab and _a

periodic

structure. We find that the periodic structure _can be crossed at

superluminal

velocities for wavenumbers in the

optical

_gap and sizes

larger

than the thickness of the system. The

corresponding

transmission coefficients _{are very} small. The

long

wavepacket

limit of the traversal time coincides with

previous analytical

results for the real component ^of the interaction time. The

imaginary

component of this time

only

affects the

change

in size of the wavepacket.

The

question

of the time spent

by

_a

particle

ⁱⁿ a

given region

of _space has received _a

great

deal of interest

recently [1-5].

The final transmission

amplitude through

_a

region

is the super-

position

of different

paths,

due to

multiple

reflections

and/or

to

partial

_waves

crossing along

different

trajectories.

Each

path corresponds

to a different traversal

time,

and _so interference effects _may

drastically change

the final traversal time. The

problem

has been

approached

^from

many different

points

of

views,

_as shown in the recent review _on the

subject by

Landauer and Martin

[I].

One _can associate the traversal time with the time

during

which _a transmitted

particle

interacts with the

region

^of

interest~

_as measured

by

_some

physical

clock which _can detect the

particle~s

_presence in the

region.

For

electrons,

this

approach

_can utilize the Larmor

precession frequency

of the

spin, produced by

_a weak

magnetic

field

acting

within the barrier

region,

_as first

proposed by

Baz'

[6j.

^The amount of

precession

clocks the characteristic

tunneling

time rr, the so-called Bfittiker-Landauer time

[7, 8].

^For

electromagnetic

_waves, we

proposed

_an

optical

clock based _on

Faraday

rotation

[9j.

The most direct method to calculate the traversal time of _a

particle through

_a

region

is to follow the behavior of _a

wavepacket

and determine the

delay

due to the structure of the

region.

In this

approach

_one has to be careful with the

interpretation

of the

results, since~

for

example,

an

emerging peak

is not

necessarily

related to the incident

peak

in _a causative way

[10].

^For more discussion of this

problem

_{see e.g.}

[llj

and references therein. Martin and

Landauer

[12]

^{studied the}

problem

of the traversal time of classical evanescent

electromagnetic

waves

by following

the behavior of _a

1N.avepacket

in _a

waveguide.

(*)Author

for correspondence

(e-mail: jrmtlfcu.um.es)

@

Les

(ditions

_de

_Physique

₁₉₉₇

654 JOURNAL DE

PHYSIQUE

I N°5

One of the most

interesting

aspects ^{of this}

problem

is the

possibility

of

achieving superluminal

velocities with evanescent waves. Nimtz's _{group was} the first _one to _measure

superluminal

velocities

[13]. They

concentrate _on

experiments

on microwave transmission

through

undersized

waveguides. Steinberg

et al. found

superluminal

velocities for

electromagnetic

_waves in the

photonic

band _gap of

multilayer

dielectric mirrors

[14j. Spielman

_et al.

[lsj

observe that the barrier traversal time of

electromagnetic wavepacket

tends to become

independent

of the barrier thickness for opaque barriers. This

phenomena closely

relates with the Hartmans theoretical

prediction

for the electron

tunneling [16j.

One controversial aspect ^{of this}

problem

is the

complex

nature of the traversal

time,

which arises both for electrons and for classical

electromagnetic

_waves. Sokolovski and Baskin

[17]

obtained _a

complex

traversal time with the

Feynman path-integral technique. They

define _a functional that _measures the time spent

by

_a

Feynman path

in _a

region

and then _sum this

quantity

_over all

possible paths

with the

weighting eis/~,

where S is the action. In the

optical clock,

the real

part

^{of the time is related} to the

angle

of

rotation,

while the

imaginary

_part is

related to the

ellipticity.

Different

phenomena

_can be associated with _one of the

components

of time. One of the aims of this _paper is to establish the

relationship

between the

components

of time and the

delay (or acceleration)

of the

peak

and the

change

of width of the transmitted

packet.

In this paper we

study

the

problem

of the time

spent by

_a classical

electromagnetic

_wave

in _a slab and in _a

layered system

_as a function of the size of the

wavepacket

when _we take

into _account the effects of

multiple

reflections. We simulate

numerically

the time evolution of

a finite size

wavepacket

that _crosses the

region

^of interest and _measure the

delay

of the

peak

of the transmitted _{wave as a} function of the size of the

original packet.

We also calculate the

change

in size of the

packet.

1. Method of Calculation

We have _a three-dimensional

layered system

^with translational

symmetry

ⁱⁿ the Y-Z

plane.

We consider N

layers

labeled I

=

I,

,

N between _two

equal

semi-infinite media with _a uniform dielectric constant ~lo. Each

layer

is characterized

by

_an index of refraction _~li. The boundaries of the I-th

layer

_are

given by

_xi and _zi+i, with _xi

" 0 and zN+i =

L,

so that the

region

of

interest

corresponds

to the interval 0 < z < L.

Later,

we will concentrate _on the

specific examples

of _a slab

(N

=

I)

and of _a

periodic arrangement

^of

layers.

A Gaussian

wavepacket

of

spatial

width _aI is incident from the left _on the

region

^of interest.

This

packet

is characterized

by

_a wavefunction of the form:

~(z, t)

=

~

C _exp

[-(k ko)~/2(/hk)~j

_exp

[ikz i~tj

dk

(1)

where C is _a normalization

constant, ko

the central

wavenumber,

_~

=

ck/no,

_c is the vacuum

speed

of

light,

and /hk

=

I/viaI

_{is the}

_spread

_{of the}

_packet

_{in the wavenumber domain.}

This wavefunction

~(z,t)

_{can represent one} of the

components

^{of either the electric} or the

magnetic

^field. The time evolution of this

wavepacket

is

governed by

Maxwell

equations

with the

appropriate boundary

conditions.

Nevertheless,

the results _are

directly applicable

to any

other classical

_wave and _to

quantum wavefunctions,

_as

long

_as

dispersion

is

neglected.

Part of the

packet

considered is transmitted and continues

traveling

toward the

right.

Its wavefunction is

given by:

~(x~ t)

=

j" _cjt(k)

jet<(ki

exp

j-(k _{ko)2 /2(/~k).2j}

_exp

jikx iwti

dk

(2)

t(k)

is the

amplitude

of transmission and

#(k)

its

phase.

As the two semi-infinite media

are

non-dispersive,

the

wavepacket

travels with _a well-defined

velocity

in both of them. We

numerically

simulate this time evolution of the

wavepacket

and calculate the time taken

by

the

packet

to cross the

region

of interest. In

particular~

we measure the average

positions

Ii and Y2 of the _square of the modulus of the

wavepacket

at two values of

t,

ti and

t2,

^{such that} the

packet

is very far _to the left of the structure at

ti

and _very far _to the

right

at t2. These

average

positions

are defined _as

Tit)

₌

/~ _{xi i~lz, t)}

i~ dz. 13)

The traversal time of the

wavepacket through

the

region

of interest is

equal

to:

r = t2 ti ~~~ ~~

~~~°

(4)

c

We also _measure the standard deviation of the transmitted

wavepacket.

We

systematically study

the behavior of the traversal time _{as a} function of the central wavenumber

ko

and of the size _aI of the

packet.

We _can obtain close

expressions

for the time

in the two extreme _cases of _very

long

and _very short

wavepackets,

_as

compared

with L. In _a

previous work,

we obtained the traversal time for _very

long wavepackets

in terms of derivatives with

respect

to

frequency

_as

_[9j:

T # -I

~~~~ j

0w _w

(5)

where _r is the

amplitude

of reflection of the

region

considered. As _we will _see, the real part of this

complex

time is associated with the traversal

time,

_as

previously defined,

and the

imaginary _part

is related to the shift in the central wavenumber

ko.

The derivative of the time with respect ^to

frequency

is associated with the distortion of the

wavepacket.

2. Results for _a Slab

Let _us first consider _a slab confined _to the

segment

⁰ < _z < L and characterized

by

an index of refraction

n. The transmission

amplitude

_t of the slab is

given by [18j:

y~2 y~2 ²

~~~

t ₌ I + ^° sin _u

exp(itfi), (6)

2nno

and the

phase

_ifi

by

~~~'~ ~~~~~

^{~~~~' ~~~}

where _u

=

wnL/c.

The reflection

amplitude

_r is

equal

to:

(y~j

~2)

~ ~~

2non

^{~~~~' ~~~}

Substituting

these

expressions

for t and r in

equation (5)

we obtain for the two time

components

in the

limiting

_case of

long wavepackets

the

following:

~~ ~

2cno

^~

4nnow

^{~~~~~ ' ~~~}

656 JOURNAL DE

PHYSIQUE

I N°5

o 10

.00

Fig.

1. _Ti and _{T2 as a} function of k for

long wavepackets crossing

a slab. The values of the parameters are L

= 10, _{n =} 2 and _no

= 1. We choose _c

= 1. The dot corresponds to k

=

81ir/80,

the square to k ₌

41ir/40

and the

triangle

to k

=

21ir/20.

The horizontal line represents the _average value of _Ti with respect to k.

and

T2 =

T~

'~° ^{~ ~} ~°

sin~

u +

~~~ _~°~

sin 2u

(10)

nno~ wnn~ cno

^~

where T

=

jtj~

is the transmission coefficient.

Throughout

the _{paper, Ti} and _T2 will represent the real and

imaginary parts

^of T in the

long wavepacket limit,

defined

by equation (5).

In

Figure

I _we represent _Ti and _T2,

given by equations (9, 10),

_{as a} function of k. The width of the slab considered _is L

= 10 and its index of refraction is

equal

to

2,

while the index of refraction outside the slab is I. We choose _c

= I. In _some wavenumber _ranges, the

oscillatory

character of the second term _on the RHS of

equations (9, 10)

results in traversal

times

significantly

smaller than the _one

corresponding

to

crossing

the slab at the group

velocity

in the medium. The three

symbols

represent ^{the value} of _Ti for each of the wavectors considered in the next

figure.

The horizontal line

corresponds

to the average value of _Ti with

respect

to k.

In the lower

part

^of

Figure

2 _we show the results of the numerical simulations of the traversal

time _versus the size of the

wavepacket

for three different values of the central

1i>avenumber,

k =

81ir/80,

41ir

/40

and 21ir

/20.

These wavenumbers _are chosen _so that sin 2u

=

1,

sin _u

= 1

and sin _u

=

0,

and _{so Ti} is _a central

value,

_a minimum and _a

maximum, respectively.

We _can check that the

long wavepacket

limit of these results

corresponds

to Ti,

given by equation (9).

In the lower

part

^of

Figure

2 _we represent the transmission coefficient for the _same situation

as in the upper

part

^{of the}

figure.

We _{can note} the

similarity

in the behaviour of the traversal time and of the transmission coefficient.

The short

pulse

limits _are

equal

to

20.8, independently

of the central wavenumber considered.

This value is the classical

crossing time, taking

into account

multiple reflection,

which for

1.00

0.90

~=

_0.80 ~

-'-

0.70

0.60

24.0

22.0 :.."

~

20.0 "'--

-_____

18.0

16.0

io ioo iooo

q

Fig.

2. Traversal time

(lower graph)

and transmission coefficient

(upper graph)

_versus the size of the

wavepacket

for _three different values ofthe central wavenumber, k ₌

81r/80 (dashed line), 41ir/40 (solid line)

and

21ir/20 (dotted line).

The values of the parameters are the _{same as} in

Figure

1.

the slab is

given by:

The transition between the two limits

I.e. _the width of the slab.

We

can _calculate the

average

of ri

exactly and

_{check that} it is equal

to

Ln/c,

sponds

to

the _classical

crossing time ithout ncluding reflection, I.e., to the

time

3. Periodic Structure

We _now consider _a

periodic arrangement

^of

layers. Layers

with index of refraction _ni and thickness

di

alternate with

layers

of index of refraction _n2 and thickness

d2

The wavenumbers in the

layers

of the first and second type are

ki

_"

wni/c

and

k2

_"

wn2/c.

Let _us call _a to the

spatial period,

_{so a}

=

di

₊

d2.

The

periodicity

of the

system

allows us to obtain

analytically

the transmission

amplitude using

the characteristic determinant method

[19j:

tN

_"

e~'~i~i cos(Nfla/2)

_I^~~~

^~~~~~~~ sin~ pa

+

~~

_~~

sin

2d2j

^~

,

(12)

sin

pa

2

ki

_k2

~

658 JOURNAL DE

PHYSIQUE

I N°5

1.oo

0.75

~ 0.50

0.25

iso

~~

100

50 F'

o ..-...:. ..."..,_,:~'~:

___~___

.,_:."... ..:... ....-..

50 ~c

""~

100

3.0 3.5 4.0 4.5 5.0

k

Fig.

3. Transmission coefficient

(upper graph)

and _Ti

(solid

line, lower

graph)

and _T2

(dotted

line, lower

graph)

_{as a} function of k for

long wavepackets crossing

_a

periodic

_system with N

= 20 interfaces.

The horizontal line corresponds to the average value of _Ti with respect to k.

where

fl plays

the role of

quasimomentum

of the

system,

^and is defined

by

cos

pa

_{= cos}

kidi

_cos

k2d2 ~~/ k2~

^{~~ sin}

kidi

sin

k2d2 (13)

When the modulus of the RHS of

equation (13)

is

greater

than

I, fl

has to be taken _as

imaginary.

This situation

corresponds

to a forbidden

frequency

window.

We concentrate in the

simplest periodic

_case, which

corresponds

to the choice

nidi

=

n2d2

This _case contains most of the

physics

of the

problem

and is also used in _most

experimental setups [14].

^{We consider} a

system

of19

layers IN

=

20)

with

alternating

^indices of refraction of 2 and

I,

and widths of 0.6 and

1.2, respectively.

In

Figure

3 _we

represent

_Ti and _T2 for this

system

as a function of k. In the

optical

_gaps, the traversal times _are

significantly

smaller than the

crossing

time _as the _vacuum

speed

of

light equal

to 16.8. The two

big

dots

correspond

to the value of _Ti of the _{two wavectors} considered in the _next

figure,

The _average of _Ti with

respect

lE+o

iE-i

lE-2

~ lE-3

E.4

E-5

ioo

~

_io

i

o-i i-o io,o ioo.o iooo.o

q

Fig.

4. Transmission coefficient

(upper graph)

and traversal time

(lower graph)

_versus the size of the wavepacket ^for two values of the central wavenumber, k

= 3.927

(solid line)

and 4.3

(dashed line).

The values of the parameters are the _{same as} in

Figure

3.

to wavenumber is

equal

to 22.8

(horizontal line),

and coincide8 with the classical

cros8ing

^time without

including multiple

reflections.

In the lower

part

of

Figure

4 _we

represent

^{the traversal} time _versus the size of the

wavepacket

for _two values of the central

wavenumber,

3.927 and 4.306. These wavenumbers

correspond

to

the center of the _gap and to a resonance. The time needed _{to cross} the

system

at the _vacuum

speed

of

light

is

equal

to 16.8. In the upper

part

of the

figure

we show the transmission coefficient for the _{same cases as} in the lower

part.

We _can note the

similarity

in the behaviour of the traversal time and of the transmission coefficient.

Again

the

long wavepacket

limit of the

traversal time coincides with _Ti, while the short

wavepacket

limit is

independent

of wavenumber and

equal

to 29. In the _case of

electromagnetic

waves

tunneling trough

undersized

waveguides analogous

results _were obtained

[20j.

We

always

have

interference,

_even for

extremely

short

wavepackets,

between

path

_corre-

sponding

to the _same traversal time. For

strictly periodic

structures _{we can never}

reach

660 JOURNAL DE

PHYSIQUE

I N°5

ioo

io

io ioo iooo

q

Fig.

5.

al a~

_as

a function of the size of the wavepacket. The values of the parameters _are the

same as in

Figure

3.

the classical

limit,

^with no interference effects. For

example

the N I

paths consisting

_on two successive reflections

land

transmission in all other

interfaces)

all add _up

constructively.

The ratio between the

heights

of the first and the second transmitted

peaks

is

equal

to

(N -1)~ _(R(

_~,

where

(R(

^{is the reflection}

coefficient,

instead of

(N I)jRj~,

result that _one would

expect

ⁱⁿ the absence of interference.

We note that the

speed

is

superluminal

for _a wide range of sizes. The minimum size of the

packets

that travel faster than

light

is about

9,

_so that the

corresponding 2aI

is very much the

same as the size of the system.

Superluminal

velocities _occur when the transmission coefficient is _very small. We _are at present

investigating

this

relationship.

In the

optical

_gap, the width of the transmitted

packet

_aT is

slightly

smaller than the width of the incident

packet

_aI.

Up

to second order in

perturbation theory,

_we obtain that this

change

in width

depends

_on the derivatives with respect to

frequency

of _Ti and _T2. As the first of these derivatives is

equal

to _zero in the centre of the _{gap, we} arrive at:

In

Figure

5 _we

plot al a(

_{as a function of the size of} the

packet.

The

long wavepacket

limit agrees very well with 11

/2)(dT2/dw),

obtained from the characteristic

determinant,

and which is

equal

to 36.7. We check that second order

perturbation theory

works

adequately

for the sizes for which _one obtains

superluminal

velocities.

4. Discussion

Our results

apply

to any

non-dispersive

_wave. The real component ^of

time,

_Ti,

corresponds

to the traversal

time,

and the main effect of the

imaginary

component, _T2, is to

change

^the size of the

wavepacket.

In the

dispersive

_case there is _a correction to the traversal time due to the

shift in wavenumber

produced by

_T2.

Acknowledgments

We would like to thank G. Nimtz for

helpful

discussion and critical

reading

of the

manuscript,

the Direcc16n General de

Investigac16n

Cientifica _y

TAcnica, project

number PB

93/l125,

and

NATO,

Collaborative Research Grant OUTR.CRG

951228,

for financial

support.

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