1. Introduction
Recently, a planar electromagnetic waveguide excited by a two-dimensional (2D) line source has been proposed [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
], where extremely high power densities can be generated and transmitted along the waveguide. The planar waveguide is a bilayer structure, where one layer is filled with air and the other filled with a specialized LHM. In the 2D case, an infinitely-long line source is placed in the waveguide to excite guided modes travelling along the waveguide without attenuation. However, such a line source is not applicable in the real-world problems. In practice, it is often necessary to employ a probe or microstrip to excite power into the waveguide. Hence it is quite important to study whether the extremely high power densities can still be generated and transmitted in the waveguide under the excitation of a three-dimensional linear electric dipole instead of the 2D point source which has been used in Ref. [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
].
In this paper, we will present detailed analysis of the field and power distributions inside the waveguide excited by an electrical dipole. We show that the guided modes excited by the dipole are quite different from those in the 2D case [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
], and a dominant mode always exists in the waveguide. Such a dominant mode is not restricted by the guidance condition and the waveguide geometry. Again, extremely high power densities are generated and transmitted along the waveguide, which however experience a slow attenuation. Numerical results validate our conclusions.
2. Theoretical analysis
We consider a planar parallel-plate waveguide, which is filled with air in Region 0 and a LHM slab in Region 1, as shown in
Fig. 1. The relative permittivity and relative permeability are denoted by
ε_{ri}
and
μ_{ri}
(
i = 0,1), respectively. The perfectly conducting boundaries of the waveguide are located at
z =
d
_{0} and
z =
d
_{2}, and the interface of the two layers lies at
z =
d
_{1}. An electric dipole directing to the
z-direction with a dipole moment
Il is placed at the origin of the Cartesian coordinate system. Here, Region 0 is occupied by air where
ε
_{r0} =
μ
_{r0} = 1, and Region 1 is a specialized LHM characterized by
where δ stands for a retardation of LHM, and γ_{ε}
and γ_{μ}
represent the permittivity loss and permeability loss, respectively. All of δ, γ_{ε}
and γ_{μ}
are small parameters. We assume the two layers to have equal thickness, i.e. d
_{0} + d
_{2} = 2d
_{1}.
From the electromagnetic theory [
6
6
.
W. C.
Chew
,
Waves and Fields in Inhomogenous Media
(
Van Nostrand Reinhold, New York
,
1990
).
], the transverse magnetic (TM) to
z waves are produced due to the excitation of the
z-directed electric dipole. Hence the magnetic field in the waveguide has only
ϕ component under the cylindrical coordinate system shown in
Fig. 1, while the electric field has both
ρ and
z components. The components of the electromagnetic fields
E_{iZ}
,
E_{iρ}
and
H_{iϕ}
can then be written as
where the subscripts iz, iρ and iϕ represent the z, ρ and ϕ components of the fields in the rth region, Il is the momentum of the dipole, and
Fig. 1. A three-dimensional linear electric dipole located in a planar waveguide filled with two-layered media.
In above equations, R
^{+} and C
^{+} represent the coefficients of forward waves in Region 0 and 1 due to the contributions of stratified media and boundaries, while R
^{-} and C
^{-} are coefficients of backward waves in the two regions:
in which kiz=ki2−kρ2 is the longitudinal wavenumber in Region i, and the denominator D is expressed as
where
Since the electric and magnetic field components mentioned above are totally different from those in the 2D case [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
], the corresponding Poynting vector will possess both
ρ and
z components, which is also different from the 2D case.
First we consider a conventional air-filled waveguide, i. e, ε
_{r1} = μ
_{r1} = 1. After simple derivations, we obtain the denominator in coefficients R
^{±} and C
^{±} as
From the above equation, we easily obtain that
and k
_{0z} = 0 are poles of E_{iz}
for the first and second orders, respectively. Physically speaking, each pole in the denominator corresponds to the propagation constant of a guided mode, and the pole k
_{0z} = 0 represents the dominant mode supported by the planar waveguide, which can be explained by the modal theory. The modes excited by the electric dipole in the air-filled waveguide are then expressed as
where A is the mode amplitude. When k
_{0z} = 0, we easily have
Obviously the dominant mode is invariant in the transverse direction, but has a 1/√
ρ decay in the
ρ direction as it travels along the waveguide. The above conclusion is quite different from the 2D case [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
], where the condition
k
_{0z} = 0 does not correspond to a physical mode supported by the planar waveguide under the excitation of an infinitely-long line source. Another important feature is that the dominant mode always exists in the planar waveguide excited by the electric dipole, without restrictions to the size of waveguide. When (
d
_{2} -
d
_{0}) <
λ
_{0}/2, only the dominant mode exists in the waveguide. When (
d
_{2} -
d
_{0}) ≥
λ
_{0}/2, however, higher modes are excited.
Now we consider the two-layer-filled waveguide. If Region 1 is a specialized lossless LHM, where
we have k
_{1z} = -k
_{0z} for the propagating waves. Therefore the denominator for above coefficients is simplified as
After simple derivations, we notice that the air-LHM waveguide has the same locations of poles as the conventional air-filled waveguide. However, the denominator is proportional to δ, which results in huge amplitudes of power densities proportioning to 1/δ
^{2}, as we will see from numerical results later.
We remark that k
_{0z} = 0 is also a pole of the second order for the waveguide, which corresponds to the dominant mode. In the dominant mode, the electromagnetic fields are unchanged in the z direction in both Regions 0 and 1. Unlike the conventional case, however, the amplitude of E
_{0z} is not equal to that of E
_{1z} since the boundary condition requires continuity of the normal components of the electric flux density D_{iz}
. Similarly, the poles k
_{0z} = nπ/(d
_{2} - d
_{0}), (n= 1,2…) determine the higher guided modes for the waveguide.
Note that the dominant mode is actually the TM
_{0} to
z mode which can never be supported under the excitation of a 2D point source [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
], and this is why the power-density distributions within the waveguide seem quite differently from those in Ref. [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
], as we will see in the numerical results.
Fig. 2. The time-averaged power density in the waveguide along the region 3000 mm ≤ y ≤ 3600 mm. (a) Conventional air-filled waveguide. (b) Air-LHM-filled waveguide with δ = 10^{-4} and γ_{ε}
= γ_{μ} = 0.
3. Numerical results
In order to show the power propagation inside the waveguide, we have computed the real part of the Poynting vector 〈S_{ρ}〉 = 12 Re(E_{z}
Hϕ*), which represents the time-averaged power density. In the yoz plane, S_{ρ}
= S_{y}
.
First we choose d
_{0} = -30 mm, d
_{1} = 30 mm, and d
_{2} = 90 mm, which ensures that only the dominant mode is guided along the waveguide. The z-directed electric dipole with the dipole moment Il = 1 mA·m is located at the origin of the coordinate system, and the operating frequency is 1 GHz. The observation region is set 10 λ away from the source, at x = 0 and y ∈ [3, 3.6] m.
Note that the power flows remain constants in the
z direction in the two regions, as expected. Hence we conclude that extremely high-power generation and transmission can still be realized under the excitation of the electric dipole. However, the power flows experience a slow spatial decay in the propagation due to the Hankel’s function, which could be clearly seen in
Fig. 2. We remark that the decay is not significant. For example, the power density observed at
y = 15 m (50 λ) is 2.9 × 10
^{8} mW/m
^{2}. Similarly, the energy conservation is not violated like the 2D case [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
] since the power flows in Region 0 and 1 have similar amplitudes but opposite directions. The net power flow along the waveguide is actually not very large.
Fig. 3. The time averaged power density in the lossy air-LHM-filled waveguide along the region 3000 mm ≤ y ≤ 3600 mm. (a) δ = -10^{-4} and γ_{ε} = γ_{μ} = 10^{-6}. (b) δ = -10^{-2}, γ_{ε} = γ_{μ} = 10^{-4}.
However, it is hard to fabricate lossless LHM in the real world, hence it is important to investigate how the loss will affect the power generation and transmission in the waveguide. As we know, when a small loss is introduced into LHM, the poles mentioned above are no longer staying on the real k_{ρ}
axis, but are located in the neighborhood of previous poles in the lossless case. As a consequence, the wavenumber of the dominant mode in the lossy LHM region becomes complex, whose real part represents the propagation constant of the mode whereas the imaginary part denotes the attenuation constant. As long as the LHM parameters are properly chosen, extremely high power can still be obtained similar to the lossless case.
We have computed the time-averaged power density in a lossy air-LHM-filled waveguide with
δ = -10
^{-4} and
γ_{ε}
=
γ_{μ}
= 10
^{-6}, as illustrated in
Fig. 3(a). As expected, extremely high power densities are observed, but their amplitude is smaller than that in
Fig. 2(b) due to the presence of loss. Even when the loss increases to
γ_{ε}
=
γ_{μ}
= 10
^{-4} and the retardation is as large as
δ = -10
^{-2}, the time-averaged power density is still 4 × 10
^{4} mW/m
^{2} at the observation point 10 λ away from the source, as shown in
Fig. 3(b). This makes the waveguide more practical. Similar to the lossless case, the amplitude of the power density is nearly linear to 1/
δ
^{2}.
When we change the waveguide dimensions to
d
_{0} = -60 mm,
d
_{1} = 60 mm, and
d
_{2} = 180 mm, two guided modes with
k
_{0z} = 0 and
k
_{0z} = π/(
d
_{2} -
d
_{0}) are allowed to propagate in the waveguide. In the case of 2D source, only one mode is permitted under the same waveguide dimensions [
5
5
.
Q.
Cheng
and
T. J.
Cui
, “
High power generation and transmission through a left-handed material
,”
Phys. Rev. B
,
72
,
113112
,
2005
. [CrossRef]
]. Since the internal fields are the sum of two guided modes with different propagation constants, the final power density distributions may behave quite differently. This can be understood since the electric field of the guided modes in Region 1 in the planar waveguide can be written as
Fig. 4. The time averaged power density in the air-filled and lossless air-LHM-filled waveguides along the region 3000 mm ≤ y ≤ 4000 mm when two guided modes exist. (a) Conventional air-filled waveguide. (b) lossless air-LHM-filled waveguide with δ = 10^{-4} and γ_{ε}
= γ_{μ}
= 0.
where A and B are coefficients. In the far distance where ρ is very large, the Hankel function behaves like an exponential function e^{ikρρ}. When two modes exist within the waveguide, the sum of two exponential functions behaves some period-like properties.
When LHM is slightly lossy, we have also computed the distributions of power flows along the planar waveguide, as shown in
Fig. 5. Here,
δ = -10
^{-4},
γ_{ε}
=
γ_{μ}
= 10
^{-6} and
δ = -10
^{-2},
γ_{ε}
=
γ_{μ}
= 10
^{-4}, respectively. The power patterns seem differently from those in
Fig. 4 since the fields in the waveguide are actually the sum of a few modes with different complex propagation constants instead of two modes with real propagation constants. However, the phenomenon of high power generation and transmission still exists as expected, and the amplitude of the power density is also proportional to 1/
δ
^{2}.
Fig. 5. The time averaged power density in the lossy air-LHM-filled waveguide along the region 3000 mm ≤ y ≤ 4000 mm when two guided modes exist. (a) δ = -10^{-4} and γ_{ε}
= γ_{μ}
= 10^{-6}. (b) δ = -10^{-2} and γ_{ε}
= γ_{μ}
= 10^{-4}.