Photonic tunneling times

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Photonic tunneling times

G. Nimtz, A. Enders, H. Spieker

To cite this version:

G. Nimtz, A. Enders, H. Spieker. Photonic tunneling times. Journal de Physique I, EDP Sciences,

1994, 4 (4), pp.565-570. �10.1051/jp1:1994160�. �jpa-00246930�

Classification PlJysics Abstracts

41.10H 73.40G

Photonic tunneling times

G.

Nirntz,

A. Enders and H.

Spieker

II.Physikalisches Institut, Universitàt _zu KôIn, Zülpicher Str. 77, ^D-50937 KôIn, Germany

(Recel

vert 9 December 1993, accepted 17 December

1993)

Abstract. We report on resonant and non-resonant photonic tunneling experiments. Resc- nant tunneling of _wave packets has revealed _a light localization, whereas trie non-resonant barrier tunneling lias corresponded ta _a zerc-time barrier _crossmg. For _a frequency-limited signal the

non-resonant tunneling transmission time yielded superlummal _group velocities, whereas _m the

case of _a resonant transition trie velocities

are much slower than _c.

1. Introduction.

The

theory

of

tunneling

time has been

continuously

studied since quantum mechanics _was founded [1, 2].

However,

reliable

experirnental

data for _a

comparison

have _trot been available until quite

recently.

In the _case of electron

tunneling

it is difficult to _{measure a} time

being

caused

by

the

tunneling

_process

only,

and

being

trot influenced

by

_any orner electron interaction.

It appears ^to be _more conclusive to

study

the

analogous photonic tunneling

of

electromagnetic

wave

packets. (Electromagnetic tunneling

modes _are known _as evanescent modes from total reflexion in

optics.)

Their interaction processes are

usually negligible

and trie _{recent progress}

m microelectronics allows _{to carry out}

experiments

with the necessary time resolution.

Presumably

results of

photonic tunneling

_are

representative

for _any type ^of wave

packet tunneling.

It has been shown that the one-dimensional quantum rnechanical

tunneling

is _m direct

analogy

with the

propagation

of

electrornagnetic

_wave

packets

in

waveguides

_{[3, 4].}

The

surprising

result of the

photonic tunneling experiments by

Enders and Nimtz has been, that

tunneling proceeds superluminal [4-6].

^This observation has been confirmed in dilferent

experiments [7, 8], however, ^the theoretical

interpretation

of the

superluminal

_group

velocity

is still under discussion [9,

10].

In this

study

_we carried out

experiments

with two dilferent

photonic barriers,

a subcut-off

waveguide,

which has _a

frequency dependent

transmission like _a

rectangular potential bottier,

and _a

multiple

quarter

wave-length

structure,

having

_a transmission function of _a

baud-stop

filter. Both

photonic

^barriers exhibit

superlurninal

transition. We shall also introduce _ex-

perimental

data of resonant and non-resonant

tunneling

times, ^{which reveals} an

essentially

dilferent behaviour.

Finally

_we ^{shall discuss} the basic dilference between _wave

propagation

566 JOURNAL DE PHYSIQUE I N°4

and tunnel mode

propagation.

Weil-defined _wave

properties

_as

velocity

and

delay

time _cease to have

physical rneaning

_m the _case of tunnel modes.

2.

Experimental.

The transmission of _wave

packets through photonic

barriers _were measured with the network

analyzer

HP 85108 and _a

sophisticated

calibration

technique.

Details of the

measuring

_pro- cedure _are

given

m references [4, 5, 11]. ^{We have} to

ernphasize,

that the rneasurements _are

asyrnptotic1-e-

non-invasive.

Any

^influence on the

experimental

results due to

signal

transmit- ter, ^detector or

geometrical

discontinuities _are ehrninated

by

the

calibration,

1-e- well-matched transition conditions _are established. For instance _a well-matched detector is

equivalent

to _a uniform and

infinitely long

transmission fine and thus establishes _an

asymptotic

detection of the

signal.

In technical _terms there is _no

standing

_{wave on} the whole transmission fine.

The

investigated

barrier type

(a)

is _a

rectangular waveguide,

which is

operated

below the cut- off

frequency.

In this

frequency

_range the

propagation

in the

waveguide corresponds formally

to the one-dimensional

rectangular

quantum mechanical

tunneling problem

of _a

potential

barrier [3, 4]. ^{The wavenumber is}

imaginary

and is determined

by

the geometry of tue

waveguide

according

to the

dispersion

relation

k _"

~~i(~/~vac)~ (~/(~~))~i~~~

_"

(~~/~)i~~ ~ii~~~

with Àvac the _vacuum

wavelength

and

b(b')

the width of the

waveguide

_as illustrated in

figure

1.

For the the cut-off

frequency

follows _u~

=

c/(2b).

The other

photonic

barrier type

(b)

is constructed

by

_a

periodic

^dielectric quarter

wavelength

structure, also shown in

figure

1. Its transmission

corresponds

to that of _a

bond-stop

filter and is

displayed

for

comparison

^{with the} transmission of the type

(a)

barrier in

figure

2.

b ~,

~L~

_Jc~

_d

la) (hi

Fig.

l. Sketch of the two investigated barrier types.

(a):

rectangular waveguide, the center part of the waveguide _is operated below, trie two adjacent guides _are operated above the cut-off frequency.

The longest transverse extension of the guide b _or b' of such _a rectangular waveguide determines the cut-off frequency, 1e. the transition from normal _wave propagation ta tunneling.

(b):

rectangular waveguide penodically loaded with dielectric quarter wave-length layers.

Resonant

tunneling

was studied with two barriers of type

(a),

which _were

separated by

a

larger waveguide,

^which was

operated

above the cut-off

frequency,

in this _way

forming

_a cavity. The

experirnental

set-up

corresponds

to _a

Fabry-Perot

interferometer with two subcut- off

waveguides

_as mirrors

[11, 12].

~

j

~ b)

fl ^-20

fi ^-40

e

-60

E~ ~)

-80

7 8 9 10 11 12

Frequency (GHz)

Fig.

2. Transmission of the two barners shown

in figure 1 _{as a} function of frequency.

3. Results.

We

begin

with

presenting

data of the

propagation

of _wave

packets through

the

(b )-type photonic

barrier. The _group

velocity

and trie

velocity

of the center of

gravity

have been measured in the experiments.

Figure

3 shows the transmission times of _{a wave}

packet through

_an empty

waveguide il

and

through

^the _same

guide

filled with dielectric quarter

wave-length layers (2).

The tunneled _wave

packet

_rnoves faster than the _wave

packet propagating through

the empty

waveguide.

^{The measured} transmission time for the

tunneling

_{was 81 ps} for trie barrier

length

of

114.2 _{mm as} follows from

figure

3. These values

yield

for the center of

gravity

and for trie _group

a

velocity

of 4.7c. The transmitted

signal

_{was a} Gaussian-like _wave

packet

with _a

frequency

band width of +0.5 GHz around _{a center}

frequency

of 8.7 GHz.

Obviously superluminal

barrier crossing takes

place

in

photonic

barriers of

(b)

and of the

previously

studied

(a)

_type [5, 6].

il-o

fi

éu

(

)0.5

~

~

@0.0

-2-10 2 3 4

Time

(ns)

Fig.

3. Measured propagation time of _a Gaussian

wave-packet (1) through

_an empty

waveguide

of 249.5 _mm length and (2) through trie _same waveguide filled with _seven plexiglass layers each 6 _mm thick (c) and separated by 12 _mm air-layers

(d),

the refractive index of the plexiglass has been 1.6

(Fig. l). The transmission data of this multiple quarter wavelength structure is shown _in figure 2.

Secondly

we bave studied _resonant

tunneling

in _a

syrnmetric

double-barrier _structure. Tue used barriers bave been of type

(a),

which _were

separated by

_a

larger waveguide forming

_a cavity with three

eigen

values _m the studied

frequency

_{range as} illustrated in the insert of

figure

4.

568 JOURNAL DE PHYSIQUE I N°4

-s ~

~

ii

j

'~

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

'Ê

~

~ -40

/Î

-60

7 8 9 10

Frequency (GHz)

ù.2 ~ O.OOfi

É _.t

'~

Î (

_ù.ùù4

~

-

~ ù i

ù.ùù2

ù-O O.OOO

7.52 7.54 7.56 7.58 7.60 7.ùù 7.ù2 7.ù4 7ù6 7.ù8

Frequency (GHz) Frequency (GHz)

_

G~ iùù

j§

40

1

~

J

Ù

~

_5ù

~ 2ù -c

o ù

7.52 7.54 756 7.58 7.6ù _{7.où 7.ù2 7.ù4 7.06 7.08}

Frequency

(GHz)

FreqUeflcY

(GHz)

Fig.

4. Transmission _us. frequency of _a double-barrier structure with three resonant transition fines below trie barriers' cut-off frequency

(insert).

Trie figures

(center)

_present the intensity _us. frequency

of the resonant fines and

(bottom)

trie phase time d~J/dw of fines II and III _{as a} function of frequency.

The resonant transitions III and II _were

analyzed,

their

frequencies being

well

separated

from the cut-off

frequency

of the barriers. We have determined from the measured fine width àw the

lime _T

=

(àw)~~

and from the

phase

derivative trie lime _{T =}

dç2/dw.

The latter represents the sc-called

phase

lime which

corresponds

to trie

velocity

of the group. The transmission times from both from trie real and from the

irnaginary

part ^{of trie wavenumber} are in agreement _as

seen from the

experimental

data

presented

in

figure

4. The resonant transitions III and II have rather

large

times of 80 _ns and 40 _ns,

respectively.

These _are about two orders of

magnitude larger

than the rneasured non-resonant transition times in trie _case of

single

barriers _or in the

case of _a double barrier at

frequencies

_away frorn the resonant transitions. This is _seen in the

graphs

of trie

phase

time at trie bottom of

figure

4.

4. Discussion.

In

previous investigations

^{Enders and Nimtz} [4-6] ^{have shown for the first time that the} group

velocity

of _{a wave}

packet

exceeds the

velocity

of

light

in

crossing

_opaque ^barriers. An _opaque barrier is defined

by

)k

L)

> 1, with k the

imaginary

_wave number. The

tunneling velocity

is calculated

by

u =

L/t,

where L is the barrier

length

and t the transmission lime of the

wave

packet

center of

gravity.

The

experimental

results have been

interpreted

in the

followmg

way, a

delay

titre is induced in front of the barriers due to the interference of the incident and trie reflected _waves. Trie hanter

crossing

^{atone takes}

place

in

zerc-time,

since trie measured

transition titre is

independent

of barrier

length

_[4-6]. The

interpretation

of the

experimental

results _{is m} agreement ^with quantum ^{mechanical studies}

by

Hartman [13] and

by

Low and Mende [14].

Group

velocities _{up to} 4.7c _were observed in

crossing

the quarter

wavelength

barrier _{as seen} in

figure

3. In this

experiment

the carrier of 8.7 GHz has carried informations within the limited

frequency

baud of + 0.5 GHz.

Recently

a

superluminal

_group

velocity

of 1.7c has been observed

by Steinberg

et al. in _an

optical

experiment _[7].

They

have measured the

optical path

dilference mduced

by

_a

photonic

barrier relative to the

optical path

in _vacuum. The experiment has been carried out ai a

single-photon intensity, however,

_m order to get a reliable

phase shift, they

had to

integrate

over many

photons.

This _average of the

optical path

^dilference was divided

by

_c to gel ^the

tunneling

lime

eventually.

In

spire

of trie used quantum detection

technique,

the

optical path

dilference itself _was measured in _a classical

procedure.

In addition _one should be _aware that

m this _very

experiment

_any medium with _a

phase velocity forger

^than c would also induce

an apparent

superluminal velocity.

In contrast, in _our microwave

experiment

trie

complex

transmission function bas been

measured,

this allows to

distinguish

between

phase

and _group

velocity

[5, 6].

Another confirmation of

superlummal tunneling

_was observed

by Ranfagni

et al.

quite recently

_[8].

They

have measured the

delay

time of _evanescent microwave

puises

between two

antenna

horns,

which

corresponds

to

signal

^velocities _up to 1.4c.

In the _case of _a double-barrier _structure short transition limes and thus

superluminal

_group

velocities have been measured

only

at

frequencies

in between the resonant transitions. Trie resonant states themselves _are characterized

by

_{a very}

large

transition time and

correspondingly by

_{a very} slow _wave

packet propagation.

This result _was obtained frorn

analyzing

both trie fine

width and the

phase dispersion. Obviously

the resonant transmission _is in agreernent ^{with the}

Krarners-Kronig

relation and the

standing

_{wave m} the

cavity

_governs the resonant

tunneling.

This behaviour

corresponds

to _a

light

localization in the cavity [15].

Eventually

_we would like to point out that in the _case of

tunneling

_or

propagation

of _evanes- cent

modes,

the concept of _wave

propagation

results in

superluminal

_group velocities and _zero- time

tunneling.

There _are three important

delay

times

describing

the

propagation

of _wave

packets

in _a dielectric rnedium: the front

delay,

the _group and technical

(frequency limited) signal delay,

and the

phase delay. They

_are

given by

the relations

tf ₌

lim(w

_~

co)ç2/w, tg

=

dç2/dw, t~

=

ç2/w

,

where _ç2 is the

phase

and _{w is} the

frequency

[16]. ^{Ail the relations} are

physical meaningful only

if the _{wave m} question has

only

a real _wave number [17]. ^{This is} not the _case for

tunnehng

modes which have _an

imaginary

_wave number and thus _no finite

phase

_ç2. In consequence of the

experiments

and in agreement ^{with the Maxwell}

equations

the

superlummal propagation

of

frequency

limited

signais by tunneling

modes is

possible

and has been evidenced in several experiments

recently [4-6,

7, 8, 12]. It has still to be

analyzed,

whether _a

superlummal

tunneled

frequency

lirnited _wave

packet

represents _a

signal

[2, 9, 10, 12,

18].

570 JOURNAL DE PHYSIQUE I N°4

Acknowledgments.

We

gratefully acknowledge

valuable discussions with J. Nitsch and E. Mielke.

References

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iii]

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1993.

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