# Relativity and the Tunneling Problem in a “Reduced” Waveguide

Authors: Eckehard W. Mielke; Miguel A. Marquina Carmona
Journal: International Journal of Optics
 (2013)

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## Abstract

Wave packets are considered as solutions of the Maxwell equations in a reduced waveguide exhibiting tunneling due to a stepwise change of the index of refraction. We discuss several concepts of “tunneling time” during the propagation of an electromagnetic pulse and analyze their compatibility with standard relativity.


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## Body

## 1. Introduction

Tunneling is often regarded as a quantum effect. Among the many recent applications is the scanning tunneling microscope exhibiting also phonon tunneling [1].However, in optics, it was discovered already by Newton as frustrated total reflection of light, compare [2]. In physics, it is largely accepted that there is some time scale [3] associated with the duration of any tunneling process [4]. In fact, it has been directly measured, for instance, with microwaves [5–8]. However, there is a lack of consensus what is the exact nature of this “tunneling time,” and a unique and simple expression [9] is still missing. Here, we will recapitulate some introductional material about its classical aspects and discuss the consequences for propagating electromagnetic waves in undersized waveguides. Our main objective is to confront them with standard relativity [10].Recently, modern versions [11] of the Michelson-Morley experiment have provided the bound of Δc/c≤10-17 for the isotropy of the velocity c of light in vacuum, one of the most stringent experimental limits in physics.The wave operator(1)□:=∇2-1c2∂2∂t2
of Jean-Baptiste le Rond d’Alembert is invariant under the general Lorentz transformations
(2)ct⟶γ(ct-v→·r→c)r→⟶r→+γ-1v2(v→·r→)v→-γv→t,
where r→:=(x,y,z) is the radius vector of an event and
(3)γ=11-v→·v→/c2
the Lorentz factor.It is quite remarkable that Riemann [12] proposed already in 1858 an invariant wave equation for the electromagnetic potential φ=cA0 in an attempt to accommodate—within his scalar electrodynamics—the 1855 experiments of Kohlrausch and Weber [13]. He estimated correctly the velocity c:=1/ε0μ0 of light in vaccum from the then known values the electromagnetic units. In 1886, Voigt [14] anticipated to some extent the invariance of the d’Alembertian (1) under what is now called a Lorentz boost (2).


## 2. Electromagnetic Plane Waves

Let us consider the electromagnetic fieldF={E→,B→} in a waveguide [15, 16]. To this end, we depart from the Maxwell equations of the Appendix which imply the wave equations
(4)(∇2-n2c2∂2∂t2){E→B→}=0,
for the electric and magnetic field. The refractive index n=εrμr is given in terms of the relative permittivity or dielectric constant εr and permeability μr of a medium.Let us restrict ourselves first to a plane wave solution written in the complex form(5){E→B→}={e→(x,y)b→(x,y)}ei(±kzz-ωt)={E→0B→0}ei(±k→·r→-ωt),
where k→ is the wave vector determining the direction of the wave propagation and the amplitudes E→0=E→0(ω) and B→0=B→0(ω) turn out to depend on the angular frequency ω=2πν. The wavelength λ=2π/|k→| is inverse to the norm k:=|k→| of the wave vector.If we substitute the Ansatz (5) into (4), the wave equation reduces to the Helmholtz equation [8]
(6)∇2{E→B→}+n2c2ω2{E→B→}=0
for the six components of the electromagnetic field F, compare [8]. The Laplace operator ∇2=∇·∇ remaining from the wave equation is only rotation invariant. Although (6) shares this feature with the time-independent Schrödinger equation, the original equations, before truncation, are distinguished by Poincaré or Galilei-invariance, respectively, which include spacetime translations.Applying the Ansatz (5), we find
(7)[-k→·k→+n2c2ω2]{E→0B→0}ei(±k→·r→-ωt)=0.
For nonvanishing fields, one can infer the dispersion relation(8)k→·k→=n2c2ω2,orkx2+ky2+kz2=n2c2ω2,
where n:=n(ω)=nphase, in general, could be frequency dependent.Like(E/c,p→) for massive fields, the frequency ω and the wave vector k→ form a 4D Lorentz vector; that is,
(9)ω⟶γ(ω-v→·k→),k→⟶k→+γ-1v2(v→·k→)v→-γωc2v→.Thus, the dispersion relation (8), like the wave equation, is indeed a Lorentz invariant in vacuum. In a medium, p→=(n/c)ℏω is Minkowski’s canonical momentum [17, 18].


### 2.1. Fourier Integral Transform

Since the wave equation is linear, a superposition of the plane wave solutions (5) is achieved via the Fourier integral
(10){E→B→}=1(2π)2∫‍dωdk3{ℰ→ℬ→}(ω,k→)ei(±k→·r→-ωt),
in order to represent more generic solutions like traveling pulses. Observe that the phase ±k→·r→-ωt is also a Lorentz invariant which implies that uniform planes waves preserve their properties in all inertial frames.Then, Maxwell’s vacuum equations(A.16) and (A.17) provide the algebraic constraints
(11)k→·ℰ→=0,k→·ℬ→=0,k→×ℰ→=±ωℬ→,k→×ℬ→=∓n2c2ωℰ→
on the Fourier coefficients ℰ→(ω,k→) and ℬ→(ω,k→).Observe that (11) are invariant under the duality transformation [19]
(12){ℰ→ℬ→}⟶-{(c2n2)k→×ℬ→-k→×ℰ→}
such that we can express the magnetic Fourier component ℬ→=k→×ℰ→/k in terms of the electric counterpart ℰ→.


## 2.1. Fourier Integral Transform

Since the wave equation is linear, a superposition of the plane wave solutions (5) is achieved via the Fourier integral
(10){E→B→}=1(2π)2∫‍dωdk3{ℰ→ℬ→}(ω,k→)ei(±k→·r→-ωt),
in order to represent more generic solutions like traveling pulses. Observe that the phase ±k→·r→-ωt is also a Lorentz invariant which implies that uniform planes waves preserve their properties in all inertial frames.Then, Maxwell’s vacuum equations(A.16) and (A.17) provide the algebraic constraints
(11)k→·ℰ→=0,k→·ℬ→=0,k→×ℰ→=±ωℬ→,k→×ℬ→=∓n2c2ωℰ→
on the Fourier coefficients ℰ→(ω,k→) and ℬ→(ω,k→).Observe that (11) are invariant under the duality transformation [19]
(12){ℰ→ℬ→}⟶-{(c2n2)k→×ℬ→-k→×ℰ→}
such that we can express the magnetic Fourier component ℬ→=k→×ℰ→/k in terms of the electric counterpart ℰ→.


## 3. Electromagnetic Waves in a Rectangular Waveguide

Let us consider a waveguide whose symmetry axis is aligned with thez coordinate and which, rectangular such that 0≤x≤a and 0≤y≤b, as sketched in Figure 1.Figure 1
Rectangular waveguide. The electric and the magnetic fieldsE→ and B→, respectively, are perpendicular to the wave vector k→=(0,0,kz).


### 3.1. Evanescent Modes due to Boundary Conditions

The boundary conditions for a waveguide with ideal conductivity at the walls are the vanishing of both, thetangential component of E→ and the normal component of B→ at the surface S, which is equivalent to
(13)n→×E→|S=0,n→·B→|S=0,
where n→ is a unit vector normal to the surface (this will break, however, the symmetry (12) under duality rotations). Since in our coordinate system, the normal vector has no z-component, that is, n·ez=nz=0, an equivalent form is
(14)Ez|S=0,n→×e→z·E→|S=0.Consequently, it is more convenient to depart from thez-components of the electromagnetic fields
(15)ez(x,y)=ℑ[c1eikxx]ℑ[c2eikyy],bz(x,y)=ℜ[c1eikxx]ℜ[c2eikyy]
as basic fields and deduce the other components from Maxwell’s equations.The boundary conditions (such idealized conditions can also be formulated by the principle of mirror images in electrodynamics) for the magnetic component are equivalent to a vanishing directional derivative(16)∂Bz(x,y)∂n|S:=nx∂Bz∂x+ny∂Bz∂y=∂∂z(n→·B→)|S+in2ωcn→×E→|S=0
at the metallic or superconducting surface S.The conditions at the extremal boundariesx=a and y=b, respectively, constrain two components of the wave vector such that the eigenvalues(17)kx=lπa,ky=mπb
are enforced, where l and m are integers. Then, the dispersion relation (8) provides the following relation
(18)kz=kz(ω)=±n(ω)cω2-ωcut2
for the component of the wave vector (wave number) kz in z direction. If we define the cut-off frequency (in Born-Infeld type generalizations of Maxwell's theory, this cut-off frequency becomes field dependent [20], a typical nonlinear effect) via
(19)ωcut:=πcn(la)2+(mb)2=cnklm,
then there results the condition
(20)ω>ωcut
for the wave number kz to be real. In the case that kz becomes imaginary, the standing waves in the xy-plane are called “evanescent” modes.Thus, for thez-component of the electric and magnetic field, we obtain the real standing waves in the xy-plane (21)ez(x,y)=E0sinlπxasinmπyb=E0ℑ(eilπx/a)ℑ(eimπy/b),bz(x,y)=B0coslπxacosmπyb=B0ℜ(eilπx/a)ℜ(eimπy/b),
respectively.The other components of the electromagnetic field can easily be determined from Maxwell’s equations as(22)ex(x,y)=1klm2(kz∂ez∂x+ω∂bz∂y),ey(x,y)=1klm2(kz∂ez∂y-ω∂bz∂x),bx(x,y)=1klm2(kz∂bz∂x-n2c2ω∂ez∂y),by(x,y)=1klm2(kz∂bz∂y+n2c2ω∂ez∂x).


### 3.2. Transversal Electric Waves

Let us restrict ourselves totransversal electric (TE) waves for which Ez=0 is implying E0=0. Then, the remaining components turn out to be
(23)ex(x,y)=-B0ωklm2mπbℜ(eilπx/a)ℑ(eimπy/b),ey(x,y)=B0ωklm2lπaℑ(eilπx/a)ℜ(eimπy/b),bx(x,y)=-B0kzklm2lπaℑ(eilπx/a)ℜ(eimπy/b),by(x,y)=-B0kzklm2mπbℜ(eilπx/a)ℑ(eimπy/b).
These components satisfy, as expected, the boundary conditions ex(x,0)=ex(x,b)=0 and ey(0,y)=ey(a,y)=0 and are perpendicular; that is, e→·b→=0.The basicdominating mode TE10 corresponds to the case l=1, m=0 and reads explicitly
(24)E→=B0cos(kzz-ωt)={0(aωπ)sin(πxa)0,B→=B0sin(kzz-ωt)={-(akzπ)sin(πxa)0cos(πxa).
The magnetic field is drawn in Figure 2, compare [16, 21].Figure 2
Projection of the magnetic component of the basic TE mode in the waveguide.Higher eigenvaluesl and m would generate different electromagnetic fields in the waveguide, the so-called higher TE modes, compare [22].


### 3.3. Effective Two-Dimensional Wave Equation

The remaining two-dimensional wave equation(25)(∂2∂z2-n2c2∂2∂t2){E→B→}=0
has the general traveling-wave solutions in z-direction:
(26){E→B→}={E→0B→0}(z∓cnt)=12π∫-∞∞‍dω{ℰ→(ω)ℬ→(ω)}ei(±kzz-ωt),
or a linear superposition of it. As indicated, they admit a Fourier transformation in terms of ω with kz=kz(ω) being complex for evanescent modes.


## 3.1. Evanescent Modes due to Boundary Conditions

The boundary conditions for a waveguide with ideal conductivity at the walls are the vanishing of both, thetangential component of E→ and the normal component of B→ at the surface S, which is equivalent to
(13)n→×E→|S=0,n→·B→|S=0,
where n→ is a unit vector normal to the surface (this will break, however, the symmetry (12) under duality rotations). Since in our coordinate system, the normal vector has no z-component, that is, n·ez=nz=0, an equivalent form is
(14)Ez|S=0,n→×e→z·E→|S=0.Consequently, it is more convenient to depart from thez-components of the electromagnetic fields
(15)ez(x,y)=ℑ[c1eikxx]ℑ[c2eikyy],bz(x,y)=ℜ[c1eikxx]ℜ[c2eikyy]
as basic fields and deduce the other components from Maxwell’s equations.The boundary conditions (such idealized conditions can also be formulated by the principle of mirror images in electrodynamics) for the magnetic component are equivalent to a vanishing directional derivative(16)∂Bz(x,y)∂n|S:=nx∂Bz∂x+ny∂Bz∂y=∂∂z(n→·B→)|S+in2ωcn→×E→|S=0
at the metallic or superconducting surface S.The conditions at the extremal boundariesx=a and y=b, respectively, constrain two components of the wave vector such that the eigenvalues(17)kx=lπa,ky=mπb
are enforced, where l and m are integers. Then, the dispersion relation (8) provides the following relation
(18)kz=kz(ω)=±n(ω)cω2-ωcut2
for the component of the wave vector (wave number) kz in z direction. If we define the cut-off frequency (in Born-Infeld type generalizations of Maxwell's theory, this cut-off frequency becomes field dependent [20], a typical nonlinear effect) via
(19)ωcut:=πcn(la)2+(mb)2=cnklm,
then there results the condition
(20)ω>ωcut
for the wave number kz to be real. In the case that kz becomes imaginary, the standing waves in the xy-plane are called “evanescent” modes.Thus, for thez-component of the electric and magnetic field, we obtain the real standing waves in the xy-plane (21)ez(x,y)=E0sinlπxasinmπyb=E0ℑ(eilπx/a)ℑ(eimπy/b),bz(x,y)=B0coslπxacosmπyb=B0ℜ(eilπx/a)ℜ(eimπy/b),
respectively.The other components of the electromagnetic field can easily be determined from Maxwell’s equations as(22)ex(x,y)=1klm2(kz∂ez∂x+ω∂bz∂y),ey(x,y)=1klm2(kz∂ez∂y-ω∂bz∂x),bx(x,y)=1klm2(kz∂bz∂x-n2c2ω∂ez∂y),by(x,y)=1klm2(kz∂bz∂y+n2c2ω∂ez∂x).


## 3.2. Transversal Electric Waves

Let us restrict ourselves totransversal electric (TE) waves for which Ez=0 is implying E0=0. Then, the remaining components turn out to be
(23)ex(x,y)=-B0ωklm2mπbℜ(eilπx/a)ℑ(eimπy/b),ey(x,y)=B0ωklm2lπaℑ(eilπx/a)ℜ(eimπy/b),bx(x,y)=-B0kzklm2lπaℑ(eilπx/a)ℜ(eimπy/b),by(x,y)=-B0kzklm2mπbℜ(eilπx/a)ℑ(eimπy/b).
These components satisfy, as expected, the boundary conditions ex(x,0)=ex(x,b)=0 and ey(0,y)=ey(a,y)=0 and are perpendicular; that is, e→·b→=0.The basicdominating mode TE10 corresponds to the case l=1, m=0 and reads explicitly
(24)E→=B0cos(kzz-ωt)={0(aωπ)sin(πxa)0,B→=B0sin(kzz-ωt)={-(akzπ)sin(πxa)0cos(πxa).
The magnetic field is drawn in Figure 2, compare [16, 21].Figure 2
Projection of the magnetic component of the basic TE mode in the waveguide.Higher eigenvaluesl and m would generate different electromagnetic fields in the waveguide, the so-called higher TE modes, compare [22].


## 3.3. Effective Two-Dimensional Wave Equation

The remaining two-dimensional wave equation(25)(∂2∂z2-n2c2∂2∂t2){E→B→}=0
has the general traveling-wave solutions in z-direction:
(26){E→B→}={E→0B→0}(z∓cnt)=12π∫-∞∞‍dω{ℰ→(ω)ℬ→(ω)}ei(±kzz-ωt),
or a linear superposition of it. As indicated, they admit a Fourier transformation in terms of ω with kz=kz(ω) being complex for evanescent modes.


## 4. Velocity Concepts for Wave Propagation

In order to unveil various velocity definitions for classical propagation [8, 23], let us consider the general wave packet
(27)Φ(r→,t)=12π∫-∞∞‍C(ω)exp{iSω(r→,t)}dω=12π∫-∞∞‍C(ω)expi{g(r→)-ωt}dω
given by a Fourier transformation of monochromatic waves, where S:=Sω(r→,t) denotes its momentary phase and g(r→)=k→·r→ for plane waves.


### 4.1. Phase Velocity

The phase velocity is the speed of a surface of constant phase, which implicitly relatesr→ and t. By definition, it satisfies
(28)dSω(r→,t):=∇S·dr→+∂S∂tdt=0.Sincek→:=-∇S and ω:=∂S/∂t define the wave vector and angular frequency, respectively, for v→ph:=dr→/dt, we obtain the relation
(29)k→·v→ph=ω
for a scalar product. This reduces to the usual definition
(30)vph=ω|∇g(r→)|≃ω|k→|,
for the absolute value which, for plane waves, reduces to the standard formula, as indicated above.Due to the dispersion relation (18) in a waveguide, the phase velocity
(31)vph=ω|k→z|=cn1-ωcut2/ω2≥cn
in z direction is, in general, larger than the velocity of light c in vacuum or in a medium with refractive index n. For ω→ωcut, it even approaches infinity. The phase index is usually defined by nph:=c/vph=ck/ω.


### 4.2. Velocity of Energy Propagation

As is well known, the Poynting vector(32)P→:=1μ0ℜ{E→×B→}
counts the energy flux of electromagnetic fields. After insertion of B→ from (11), one obtains for waves the following identity:
(33)P→=1μ0ωℜ{E→×(k→×E→)}=1μ0ωℜ{(E→·E→)k→-(E→·k→)E→}=1μ0ω|E→|2k→.
(In view of (29), there results also the scalar relation P→·v→ph=|E→|/μ0).Then, for nondispersive media, the energy velocity can be defined by(34)v→energy=P→ρem.
The energy density of electromagnetic waves is
(35)ρem=12ϵ0ℜ{E→2+c2B→2}.
Then, it can be shown [24] that its absolute value satisfies the inequality
(36)|v→energy|≤cn.
However, physically, this velocity is not always measurable and, in view of (33), is not invariant under the Lorentz transformations (9).


### 4.3. Group Velocity

After superposition, the resulting localized wave packet has an envelope built from all amplitudes which determines thegroup velocity satisfying
(37)dk→·v→group=dω.
Since it represents the first term of a Taylor series for the modulation velocity, it can be rewritten as
(38)v→group=∇kω(k).
In the case gω(r→)=k→·r→, this reduces to the familiar formula
(39)vgroup=dωdk|ω-=cn(ω)+ωdn(ω)/dω.
Here we included the case of a frequency-dependent refractive index n=n(ω). This velocity is related to the speed of the maximum of the wave packet. The case that ω=kc/n provides another useful formula
(40)vgroup=vph[1-kndndk]=cn+λcn2dndλ.
Depending on the slope dn/dω of the frequency-dependent (using Lorentz's dispersion relation n2(ω)=1+Ne2/ε0me[∑j‍νj/(ω0j2-ω2)] for the frequency-dependent index of refraction, one finds the alternative formula vgroup=c/(1+Ne2/2ε0meω2), this lends itself to the relation ngroup:=c/vgroup=n(ν)+νdn(ν)/dν for the group index) n=n(ω) refraction index, for anomalous dispersion [25], rather low, “superluminal,” or even negative [26] group velocities can occur. Partially, this has been confirmed by recent measurements [27] in Erbium-doped optical fibers.In the case of a waveguide, we can use (18), calculate first dk/dω, and then form its inverse. The result is
(41)vgroup=vph1-ωcut2/ω21+(1-ωcut2/ω2)dlnn/dlnω=cn1-ωcut2/ω21+(1-ωcut2/ω2)dlnn/dlnω≤cn.
For ω>ωcut and normal dispersion, this result is in agreement with the theory of relativity. Moreover, in the limit ω→ωcut, it approaches zero as expected.In view of (31), the product
(42)vphasevgroup=c2n2[1+(1-ωcut2/ω2)dlnn/dlnω]⟶c2n2≤c2
has a constant limit smaller than the velocity of light c squared in vacuum for a frequency-independent refraction index n.In general, it is difficult to distinguish in a superposition of waves what is the carrier and what is the modulation in a specific waveform. This can already be seen in the simple example of the superposition of two sinusoidal wave trains with slightly different frequenciesω±Δω and wave numbers k±Δk: after applying addition theorems for trigonometric functions, the superposed wave (43)2cos(Δkx-Δωt)cos(kx-ωt)
formally appears to exhibit two “phase velocities”: one is ω/k for the carrier and the other the group velocity Δω/Δk of the “envelope wave.” However, such an identification is not always unambiguous.


### 4.4. Delay Time

Thedelay time of a wave packet or group propagating in a medium is usually [28, 29] defined by
(44)τgroup:=∂S∂ω=ℏ∂S∂E,
where the equivalent formula is written for photons of energy E=ℏω, compare [30].Observe that the quotient of the traveled distance divided by the delay time returns back the group velocity; that is,(45)vgroup=zτgroup=vph1-ωcut2/ω21+(1-ωcut2/ω2)dlnn/dlnω.The delay time was already discussed by MacColl [31] in 1932. However, there is still not really a consensus about the time a signal spends during tunneling through a barrier, in particular in the framework of quantum mechanics [3, 9, 32, 33] and for single photons [34, 35].


## 4.1. Phase Velocity

The phase velocity is the speed of a surface of constant phase, which implicitly relatesr→ and t. By definition, it satisfies
(28)dSω(r→,t):=∇S·dr→+∂S∂tdt=0.Sincek→:=-∇S and ω:=∂S/∂t define the wave vector and angular frequency, respectively, for v→ph:=dr→/dt, we obtain the relation
(29)k→·v→ph=ω
for a scalar product. This reduces to the usual definition
(30)vph=ω|∇g(r→)|≃ω|k→|,
for the absolute value which, for plane waves, reduces to the standard formula, as indicated above.Due to the dispersion relation (18) in a waveguide, the phase velocity
(31)vph=ω|k→z|=cn1-ωcut2/ω2≥cn
in z direction is, in general, larger than the velocity of light c in vacuum or in a medium with refractive index n. For ω→ωcut, it even approaches infinity. The phase index is usually defined by nph:=c/vph=ck/ω.


## 4.2. Velocity of Energy Propagation

As is well known, the Poynting vector(32)P→:=1μ0ℜ{E→×B→}
counts the energy flux of electromagnetic fields. After insertion of B→ from (11), one obtains for waves the following identity:
(33)P→=1μ0ωℜ{E→×(k→×E→)}=1μ0ωℜ{(E→·E→)k→-(E→·k→)E→}=1μ0ω|E→|2k→.
(In view of (29), there results also the scalar relation P→·v→ph=|E→|/μ0).Then, for nondispersive media, the energy velocity can be defined by(34)v→energy=P→ρem.
The energy density of electromagnetic waves is
(35)ρem=12ϵ0ℜ{E→2+c2B→2}.
Then, it can be shown [24] that its absolute value satisfies the inequality
(36)|v→energy|≤cn.
However, physically, this velocity is not always measurable and, in view of (33), is not invariant under the Lorentz transformations (9).


## 4.3. Group Velocity

After superposition, the resulting localized wave packet has an envelope built from all amplitudes which determines thegroup velocity satisfying
(37)dk→·v→group=dω.
Since it represents the first term of a Taylor series for the modulation velocity, it can be rewritten as
(38)v→group=∇kω(k).
In the case gω(r→)=k→·r→, this reduces to the familiar formula
(39)vgroup=dωdk|ω-=cn(ω)+ωdn(ω)/dω.
Here we included the case of a frequency-dependent refractive index n=n(ω). This velocity is related to the speed of the maximum of the wave packet. The case that ω=kc/n provides another useful formula
(40)vgroup=vph[1-kndndk]=cn+λcn2dndλ.
Depending on the slope dn/dω of the frequency-dependent (using Lorentz's dispersion relation n2(ω)=1+Ne2/ε0me[∑j‍νj/(ω0j2-ω2)] for the frequency-dependent index of refraction, one finds the alternative formula vgroup=c/(1+Ne2/2ε0meω2), this lends itself to the relation ngroup:=c/vgroup=n(ν)+νdn(ν)/dν for the group index) n=n(ω) refraction index, for anomalous dispersion [25], rather low, “superluminal,” or even negative [26] group velocities can occur. Partially, this has been confirmed by recent measurements [27] in Erbium-doped optical fibers.In the case of a waveguide, we can use (18), calculate first dk/dω, and then form its inverse. The result is
(41)vgroup=vph1-ωcut2/ω21+(1-ωcut2/ω2)dlnn/dlnω=cn1-ωcut2/ω21+(1-ωcut2/ω2)dlnn/dlnω≤cn.
For ω>ωcut and normal dispersion, this result is in agreement with the theory of relativity. Moreover, in the limit ω→ωcut, it approaches zero as expected.In view of (31), the product
(42)vphasevgroup=c2n2[1+(1-ωcut2/ω2)dlnn/dlnω]⟶c2n2≤c2
has a constant limit smaller than the velocity of light c squared in vacuum for a frequency-independent refraction index n.In general, it is difficult to distinguish in a superposition of waves what is the carrier and what is the modulation in a specific waveform. This can already be seen in the simple example of the superposition of two sinusoidal wave trains with slightly different frequenciesω±Δω and wave numbers k±Δk: after applying addition theorems for trigonometric functions, the superposed wave (43)2cos(Δkx-Δωt)cos(kx-ωt)
formally appears to exhibit two “phase velocities”: one is ω/k for the carrier and the other the group velocity Δω/Δk of the “envelope wave.” However, such an identification is not always unambiguous.


## 4.4. Delay Time

Thedelay time of a wave packet or group propagating in a medium is usually [28, 29] defined by
(44)τgroup:=∂S∂ω=ℏ∂S∂E,
where the equivalent formula is written for photons of energy E=ℏω, compare [30].Observe that the quotient of the traveled distance divided by the delay time returns back the group velocity; that is,(45)vgroup=zτgroup=vph1-ωcut2/ω21+(1-ωcut2/ω2)dlnn/dlnω.The delay time was already discussed by MacColl [31] in 1932. However, there is still not really a consensus about the time a signal spends during tunneling through a barrier, in particular in the framework of quantum mechanics [3, 9, 32, 33] and for single photons [34, 35].


## 5. Scattering at Dielectric Interfaces

The refractive indexn may change discontinuously at two interfaces located at z=0 and z=L, corresponding to the insertion of a block of dielectric material into the waveguide. Then,
(46)n(z)={nz<0n~0≤z≤Lnz>0
as sketched in Figure 3. In vacuum, we have n=1.Figure 3
“Barrier” of a dielectric material withn~>n.As we have seen, such a barrier can effectively also be archived by a reduction of the size of the guide at some portion of lengthL in the laboratory frame. According to (18), in the case of a reduced waveguide, one has kz=n~ω/c, where the effective refraction index is identified via n~=n1-ωcut2/ω2.For plane waves, the Fresnel coefficients(47)r=n-n~n+n~,r=-r′,t=2nn~n+n~,t=t′
of transmission Teikz and reflection Re-ikz can be formally [36] rearranged in a Heisenberg-Wheeler [37] type scattering matrix(48)S={rt′tr′}.Then, in concordance with energy conservation, the coefficients of transmission and reflection obey(49)T:=|t|2=4k12k224k12k22+(k12-k22)2sin2(k2L),R:=|r|2=T-1,
where k1 and k2 are the wave numbers outside the barrier.Figure4 indicates the behavior of the wave functions in the different regions of such a barrier, see also [38].Figure 4
Tunneling probability for a harmonic wave in a potential “barrier” withn~>n and length L.To some extent, the refraction index plays the analogous role of a potentialV(z) in quantum mechanics: Then the transmission coefficient decays exponentially within the barrier; that is, the WKB approximation [4, 39] of the Schrödinger equation yields
(50)T∝exp(-2ℏ[2m(V0-E)L]),
in the case constant kinetic energy E<V0. However, the applicability of this analogy is quite limited, since the time-dependent Schrödinger equation is only Galilei invariant.


### 5.1. Step-Modulated Incident Wave

As an idealization to a physically more realistic form of transmitting a signal, let us now consider awave packet exhibiting initially a step-like distribution
(51){E→B→}={e→0(x,y)b→0(x,y)}Θ(t~)ℑexp{i(kzz∓ωt)}
in the z direction of propagation, where t~:=kzz∓ωt. As is well known, for a unit square step function
(52)Θ(t~):={1|t~|<π0|t~|>π,
the Fourier transform or its absolute value
(53)C(ω)=22π|sin(πω)ω|
involves an infinite frequency spectrum.Real pulses, however, are limited in the frequency band [40]. Then, the inverse Fourier transform, truncated to the first N terms in its corresponding series, would not exactly return the discontinuous step function. This is due to the Wilbraham-Gibbs phenomenon [41–43] which, mathematically, implies that the Fourier transform converges nonuniformly and, at a discontinuity, merely to its arithmetic mean. This phenomenon can also be understood via a temporal convolution of the Fourier transform with the step function Θ(ω) for a frequency band limited by |ω|<Nω. Consequently, physical signals are not really step like but appear rather “rounded off.”When considering a propagating “round” distribution in spacetime, like a Gaussiang(t~)=exp{-t~2/σ2}, instead, the response is a related distribution with half-with 〈Δω〉=2/σ in frequency space. Then, the estimate
(54)〈Δt〉〈Δω〉≃2π
for the classical Fourier transforms [28] is satisfied.In optics, this is known as a classical analogue ofHeisenberg’s uncertainty relation. In contradistinction to well-known relation for the expectation value of momentum and position squared in quantum mechanics, the relation (54) does not involve Planck’s (reduced) constant ℏ. For step-like modulated Klein-Gordon waves, it has been shown [44] that the full Fourier transform leads to vanishing waves in the superluminal region, overlooking, however, a possible uncertainty due to the Gibbs phenomenon. Similarly, the recent reanalysis [45] of Sommerfeld and Brillouin forerunners may also depend on limitations from a finite frequency band.It is also important to note that the evolution of wave packets is not confined to a simple displacement. Its amplitude squared also changes with time such that it exhibits a “dispersion” or pulse-reshaping [35] which is minimal for t=0.


## 5.1. Step-Modulated Incident Wave

As an idealization to a physically more realistic form of transmitting a signal, let us now consider awave packet exhibiting initially a step-like distribution
(51){E→B→}={e→0(x,y)b→0(x,y)}Θ(t~)ℑexp{i(kzz∓ωt)}
in the z direction of propagation, where t~:=kzz∓ωt. As is well known, for a unit square step function
(52)Θ(t~):={1|t~|<π0|t~|>π,
the Fourier transform or its absolute value
(53)C(ω)=22π|sin(πω)ω|
involves an infinite frequency spectrum.Real pulses, however, are limited in the frequency band [40]. Then, the inverse Fourier transform, truncated to the first N terms in its corresponding series, would not exactly return the discontinuous step function. This is due to the Wilbraham-Gibbs phenomenon [41–43] which, mathematically, implies that the Fourier transform converges nonuniformly and, at a discontinuity, merely to its arithmetic mean. This phenomenon can also be understood via a temporal convolution of the Fourier transform with the step function Θ(ω) for a frequency band limited by |ω|<Nω. Consequently, physical signals are not really step like but appear rather “rounded off.”When considering a propagating “round” distribution in spacetime, like a Gaussiang(t~)=exp{-t~2/σ2}, instead, the response is a related distribution with half-with 〈Δω〉=2/σ in frequency space. Then, the estimate
(54)〈Δt〉〈Δω〉≃2π
for the classical Fourier transforms [28] is satisfied.In optics, this is known as a classical analogue ofHeisenberg’s uncertainty relation. In contradistinction to well-known relation for the expectation value of momentum and position squared in quantum mechanics, the relation (54) does not involve Planck’s (reduced) constant ℏ. For step-like modulated Klein-Gordon waves, it has been shown [44] that the full Fourier transform leads to vanishing waves in the superluminal region, overlooking, however, a possible uncertainty due to the Gibbs phenomenon. Similarly, the recent reanalysis [45] of Sommerfeld and Brillouin forerunners may also depend on limitations from a finite frequency band.It is also important to note that the evolution of wave packets is not confined to a simple displacement. Its amplitude squared also changes with time such that it exhibits a “dispersion” or pulse-reshaping [35] which is minimal for t=0.


## 6. Discussion: Relativity for a Rescue?

Microwaves obey the relativistic-invariant wave equation (4). Consequently, the phase of propagating solutions is not affected by a Lorentz boost (2) where v→ is the (phase) velocity of the wave. However, due to the relative definition of simultaneity, there are no rigid bodies in relativity. Consequently, the laboratory length L of the barrier or undersized waveguide, compare Figure 5 adapted from [30], gets changed via the Lorentz-FitzGerald contraction(55)L⟶L′=L1-v2c2
in a frame moving [46] with relative velocity v.Figure 5
Minkowski diagram of the tunneling of photons through a potential “barrier” of lengthL measured in the laboratory frame.Actually, due to the finite velocityc of light in vacuum, aberration will induce an apparent Penrose-Terrell rotation of an extended body, such as a reduced wave guide, as first pointed out in 1924 by Lampa [47]. More important here is the longitudinal Doppler shift of the frequencies ω of microwaves due to the Doppler factor
(56)D=(1-β)(1+β)
as a result of the Lorentz transformation (9) in phase space. On top of that there will occur a drastic decrease of the intensity proportional to D5, the so-called searchlight effect [48, 49] known from relativistic visualization.Consequently, viewed from the comoving frame of the photons withv→c/n, the barrier length becomes contracted in vacuum even to L′→0. Since the transmission coefficient (50) decays exponentially with the barrier length, the matching conditions at the boundaries of the barrier remain consistent even when viewed from the “photon frame.”Thus, already from this rather classical point of view one would expect that tunneling of evanescent modes would have zero dwell time, compare [30, 50, 51], in accordance with relativity.When considering propagating waves within the waveguide in an equivalent, butnondiagonal, form of the Minkowski metric, an event horizon occurs at v2=c2, which could induce an Unruh effect of particle creation as the equivalent of tunneling [52, 53]. This may open up a way to measure [54] already in a waveguide the analogue of the Bekenstein-Hawking effect usually associated with the occurrence of a horizon for black holes. When perturbating the Riemannian metric of such spacetime solutions, one has to be careful in avoiding [55] the Gibbs phenomenon.Tunneling in quasi macroscopic systems like SQUIDs depends also on friction [56], whereas flexural modes of graphene or carbon nanotubes [57] exhibit a double-well potential.



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*Source: 947068-2013-11-18.xml*