This paper was awarded in the III International Competition (1994/95) "First Step to Nobel Prize in Physics" and published in the competition proceedings (

The quantitative aspects of the tunneling of a Gaussian pulse through a barrier are discussed. The salient features are: the tunneling speed is generally greater than the vacuum speed of light. However causality is not violated. The tunneling time does not depend on the thickness of the forbidden zone. A lateral displacement of the transmitted pulse is observed as is known from total reflexion. The propagation vector of the transmitted pulse deviates from that of the incident pulse.

PACS numbers: 42.25.Bs, 42.25.Gy, 42.65.Re

**1. Introduction**

The velocity of a light pulse in a dielectric medium without dispersion is given by the speed of light in that medium. However, at the boundaries of such media some interesting effects occur especially under conditions of total reflection. If the boundary at which total reflection occurs does not represent an infinitely extended barrier, partial tunneling through the barrier takes place. Experiments with multilayered mirrors and other barriers [1] which reflect almost all the light have shown that transmission of light appears to occur at superluminal speeds and that transmission speeds do not depend on the thickness of the barrier.

In this paper we discuss the effects for a special kind of barrier which is hit by a Gaussian pulse.

**2. The arrangement of the barrier**

As a barrier we define a slab of vacuum which separates two dielectric
media with a given refractive index *n*>1. The critical angle of incidence
is given by

Light pulses with an angle of incidence greater than this critical angle are almost completely reflected. However, there is a part of the electromagnetic field which can tunnel through the slab if its thickness is of the order of magnitude of the average wavelength of the pulse. It is this transmitted part of the pulse which we are interested in. Figure 1 outlines the arrangement of the boundary for the incoming pulse, resulting in a reflected and transmitted component.

**3. The description of the pulse**

For the sake of simplicity we take a Gaussian pulse as can be provided by a
mode-locked laser. For our investigation we define a two-dimensional
coordinate system in the plane of incidence (the *x*,*z*-plane in Fig. 1).
Let us first introduce the two unit vectors and ;
being parallel to the propagation direction of the pulse,
being perpendicular to . With the angle of incidence
we obtain

The frequency of the laser is , which leads to the wave vector

We choose the coordinate system in such a way that, at the time *t*=0, the
pulse maximum is located at the origin, i.e., the electric field at *t*=0
is given by

where

In Eq. (4) the phase of the electric field at a specific point is determined by . The field magnitude is a Gaussian distribution with the parameters

where

The integral (6) is the superposition of plane waves which propagate in
time. Thus, the complete description of the pulse in time and space is
obtained as

where

The Fourier transform (8) of the pulse allows us to calculate for each plane wave the corresponding reflected and transmitted waves and to integrate the contributions of all wave vectors.

**4. A plane wave at the barrier**

We will now investigate what happens to a plane wave which hits the
barrier. denotes the wave vector, is the angle of
incidence. The plane wave can be written in the form

According to Snellius' law the angle of refraction of the wave transmitted into the vacuum slab is

This angle is only real as long as is smaller than the critical angle (1). The transmitted wave is described by

where is the position of the boundary. The wave vector is . The coefficient

where the indices p and s indicate the polarization of the electric field parallel and perpendicular, respectively, to the plane of incidence.

Let us examine what the evanescent wave looks like if the angle of
incidence is greater than the critical angle. We use (11) to get
an expression for

For the transmitted wave (12), we then obtain

where we have chosen the negative sign in (14) because otherwise the amplitude of the transmitted wave would increase exponentially. As we can see, the electric field in the slab is an evanescent wave which propagates parallel to the

The evanescent wave in the slab produces a transmitted wave at the second
boundary

We will consider only parallel polarized waves. The description of the transmitted wave is obtained from (10), (13), (15), (16) and (17)

and with (14)

If we compare the formulae (18) and (10) for a selected value of

Let us transform the amplitude in (18):

where

and

Finally, we have

**5. The transmitted pulse**

Let us return to the Gaussian pulse. In order to obtain the description of
the transmitted pulse we have to replace the expressions for the plane
waves in (8) by (21)

The dependence of
is given by (see Fig. 2)

It is useful to recall the relations

which can be obtained from Fig. 2. is the angle of incidence corresponding to . The main contributions of the integral (22) come from values

In order to see the influence of the new terms in the exponential function
we need the following expansions:

where

Terms of higher order in

For the real part we get the approximation

with

and

The imaginary part is obtained as

with

and

Replacing the above approximations in (25) yields

with

Introducing the new integration variables

transfers (30) into

This is the description of a Gaussian pulse with new parameters. Let us determine them by the two relations

The resulting values are only approximations in the first order of

The main frequency of the pulse is also slightly modified, i.e. there is a small red shift caused by the tunneling

The latter increases with the angle of incidence and the thickness of the slab. The shorter the incoming pulse the greater the frequency shift, the latter depending on the inverse square of the pulse duration.

Let us find the maximum of the transmitted Gaussian wave packet at a
specific time *t*. In a homogeneous medium without dispersion the maximum
of a Gaussian pulse travels at a constant speed given by the vacuum speed
of light divided by the refractive index of the medium. Its amplitude
decays with time because of the diffraction of the pulse. After the pulse
travelled a distance , its width can be derived as

where

The transmitted wave propagates in the direction of the new wave vector . The physical interpretation of this directional change is the following. There are wave components of the incoming pulse that have propagation directions which are more favorable for transmission through the slab than along the main direction of the pulse. These preferentially transmitted components decay less in the slab than the other components of the pulse. These pulse components having the smaller angles of incidence constitute the main part of the transmitted pulse. This explains why the transmitted pulse leaves the slab under an angle slightly smaller than the initial angle of incidence.

Where and when does the transmitted pulse occur on the opposite side of the
slab? Equation (32) describes the pulse after transmission through the
slab. By way of extrapolation to the time *t*=0, we may define a virtual
origin of the pulse, which is shifted by , with
respect to the true origin (*x*=0, *z*=0). From this virtual origin and the
propagation vector we obtain the exit position (,
) for the transmitted pulse

The first term is the

The first term denotes the time when the incoming pulse reaches the slab. The second expression is due to the shorter distance which pulse components with a smaller angle of incidence have to travel. The last term, which is rather small, denotes the additional time which is required for the lateral displacement of the pulse. If the pulse is not extremely wide (small

Let us consider the tunneling time. There are some difficulties to define a
tunneling time. Experimentally it is not possible to detect directly the
time point when the pulse leaves the slab, but the pulse must be observed
somewhere in the adjacent medium behind the slab. The time needed by the
pulse to get from its origin to the point of detection can be measured. The
question is what we should compare it to. We could do the same experiment
without a slab. However, as we have seen the transmitted pulse has a
slightly different direction than the original pulse. Thus, an identical
but undisturbed pulse will not be detected at the same place as the
transmitted pulse. Thus, the two paths of the pulses cannot be directly
compared. It is much better to take as reference a pulse which travels
through the homogeneous medium with the same direction as the transmitted
pulse (see Fig. 3). In this case the two paths are parallel and differ only
in the small lateral shift caused by the slab. Let us define the tunneling
time as the difference between the time point when the maximum of the
transmitted pulse appears on the far side of the slab and the time point
when the undisturbed pulse hits the slab. This yields

This value is independent of the thickness of the slab!

Fig.3.The arrangement of the experiment to measure the time differences between tunneled and undisturbed pulses.

If we calculate the difference between the time needed by a pulse to reach
the detector (Fig. 3) either with or without a slab in between we obtain

This result is quite interesting. It shows that the tunneling happens almost instantaneously. The time required for the tunneling corresponds to the time which the evanescent field needs to cover the lateral shift. Had we defined the tunneling time as the difference between the time points of the incoming and outgoing pulse maxima, we would even have obtained a negative value.

The question is whether these results violate causality. Apparently the tunneling time for the pulse maximum is superluminal. However, it must be emphasized that it is only the maximum which appears to travel at this speed. Actually the pulse is reshaped in the slab and comes out in a different form. Furthermore, the energy distribution of Gaussian pulses is not limited in space. Therefore, we cannot apply the principle of causality in its simplest form. Only if there was a distinct front of the pulse one could state that at any point no signal can be detected before the pulse front propagating with the vacuum speed of light would reach it.

**6. Visualization of tunneling**

Fig.5.Development of a pulse reflected at and transmitted through a barrier (viewed along thez-axis).

Fig.6.Development of a pulse reflected at and transmitted through a barrier (viewed along thex-axis).

The following pictures are not based on the above approximations, but take
into account multiple pulse reflections. They show the development of a
tunneling pulse in time. The first series (Fig. 4) gives an overview: the
incident pulse approaches from behind (*z*<0, *x*>0). In the first picture
(Fig. 4a) the pulse maximum is expected to be still in front of the slab
which is located between *z*=0 and *z*=3. The *z* coordinate is scaled in
units of vacuum wavelengths of the laser. As the pulse is very close to the
slab, parts of the incident and reflected pulse interfere. This causes the
disturbances and the high values of the amplitude in the vicinity of the
slab. Actually one would find standing waves in front of the slab (*z*<0);
however, this fine structure is not resolved. As can be
seen, the field magnitude decreases very rapidly in the slab.
The maximum of the transmitted pulse already appears in the second picture
(Fig. 4b) when the calculated position of the maximum of the incident pulse
has not yet reached the slab. The last two figures (Fig. 4c, d) show the
further development. Apart from the transmitted Gaussian pulse we can see a
second small pulse. Probably it is due to the multiple reflections or to
higher order terms which we did not take into account in our
approximations. The following two series (Fig. 5a-d and 6a-d) show the
pulse at the same time points, but in views perpendicular to the *z*-axis
and *x*-axis, respectively. In these views the two main effects related to
tunneling are nicely born out. In the series of Fig. 5 the backward shift
can be observed, whereas in the series of Fig. 6 the fact that tunneling
occurs faster than reflection is evident. The maximum of the transmitted
pulse already leaves the slab before the incident pulse hits the slab. In
Fig. 6d, the transmitted pulse is farther away from the slab than the
reflected one.

**Acknowledgment**

I would like to thank Prof. J. Mostowski (Polish Academy of Sciences, Warsaw) for introducing me to this problem. I am also very grateful to Dr. Ch. Fattinger (F. Hoffmann-LaRoche Ltd, Basel) for his interest and many helpful comments and discussions.

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