The organization would be mainly divided into six parts in this thesis. Besides basic theory of metamaterials described in Chapter 2, the following chapters will be introduced the detail about my research. Three topics concerning the improvement will be presented in Chapter 3 , Chapter 4 and Chapter 5. A new structure with SRRs and wires locating on the same plane will be discussed in Chapter 3. In Chapter 4, the idea of double resonant frequency is proposed while trying to compensate the loss of metamaterials. An additional chapter showing some experiment results of metamaterials at 1550nm infrared region will be discovered and analyzed in Chapter 5. Finally, the conclusion and some future works will be included in Chapter 6.
Chapter 2
Basic Theory of
Left-Handed Materials
Veselago, in a paper published in 1968 [1], pondered the consequences for electromagnetic waves interacting with a hypothetical material for which both the electrical permittivity ε and the magnetic permeability µ were simultaneously negative. Because negative permittivity and permeability have not ever been demonstrated to exist in natural materials or compounds, Veselago wondered whether this material was just a fantasy or perhaps had a more fundamental origin. He concluded that not only should this material be possible, but if ever found, it would exhibit remarkable properties unlike any other known materials and bring huge impact to all electromagnetic phenomena.
Since the vectors EK , HK
, and kK
of a plane wave in such material form a left-haneded set, Veselago referred to the material as left-handed materials (LHMs).
This material is also called negative refraction index material or metamaterials due to the negative property revealing in it.
Veselago’s concept was not realized until 1996, while Pendry purposed a periodic thin wire structure which exhibits negative permittivity below the plasma frequency [6]. Three years later, a nonmagnetic split ring resonator (SRR) performing negative permeability below the magnetic plasma frequency was discovered by Pendry [3]. In 2001, Smith combined periodic wires and SRR structure, and successfully demonstrated the phenomena of negative refraction by measuring the
refraction angle of composite LHMs which gave the evidence of negative refraction index [4]. Since then, the research related to LHMs has inspired great interest. In this chapter, we will review the fundamental electromagnetic theory and experimental demonstration of LHMs.
2.1 Veselago’s Idea
To understand Veselago’s idea, we have to start from Maxewell equation and the constitutive relations first.
It can be understood clearly from the equation that if both permittivity and permeability are positive, the vectors EK
, HK
, and kK
form a right-handed set. On the contrary, if the sign of permittivity
ε
and permeabilityµ
changes from positive to negative, then EK, HK
, and kK
will establish a left-handed triplet of vectors. In this condition, Eq.2.3 can be modified as:
k E H
which is only related to electrical and magnetic fields, it can also be described in the left-handed concept. Fig. 2.1 (a) displays the relation of Eq.2.5 in a conventional
medium where energy flux SK
is parallel to the wave vector while phase and group velocity coincide with each other.
(a) (b)
Figure 2.1: Poynting vector in right-handed and left-handed media
In LHMs, however, the Poynting vector is anti-parallel to the wave vector which shows that the phase and group velocity are in opposite directions. In other words, the term of LHMs is equivalent to the term “material with negative group velocity.”
Meanwhile, since the energy flux SK
is opposite to the phase velocity, the wave front would move toward energy source and is opposite to the propagating direction.
Hence some phenomena such as Doppler effect and Vavilov-Cerenkov effect will be reversed.
Now, let us consider the consequence that will happen on the interface between LHM and RHM. First, the boundary condition of electromagnetic wave must be considered. According to Eq.2.5, the tangential component of E and H are unaffected.
However, the normal components of EK
and HK
, with
ε
< 0 andµ
< 0, will undergo a change of sign at the interface between a RHMs and a LHMs as shown in Fig. 2.2.E
t1E
t2E
n1E
n2H
t1H
t2H
n2H
n1RHM LHM
Figure 2.2: Boundary condition at the interface between a RHM and a LHM.
Since the relation between RHMs and LHMs is different from conventional condition, Snell’s Law can be modified in a special view. That is, while the wave transmitting through the boundary of RHMs and LHMs, it will introduce a negative refraction angle which can lead to a negative refraction index in the LHMs according to Snell’s Law : n1sinθ1=n2sinθ2.
Figure 2.3: The refraction angle showing the property of LHM.
Furthermore, the mathematic calculation of n= ε µ is another explanation to derive negative refraction index in LHMs. It should be more careful in taking the square root because
ε
andµ
are analytic functions whose values are generally complex. There is an ambiguity in the sign of the square root that is solved by aproper analysis. Hence, by taking ε =i ε and µ =i µ , we can get a negative sign for
n
in LHMs. The step of taking the square root of eitherε
orµ
alone must have a positive imaginary part is a necessary one for passive material.
In summary, the negative refraction index comes from the combination of negative permittivity and negative permeability. Special phenomena, such as negative refraction angle, reverse Doppler effect and revised boundary condition, can be predicted basing on the derivation. Practical implementation of negative permittivity and negative permeability will be elucidated in next two sections.
2.2 Wire Structure of Negative Permittivity
In this section, the effect of metallic thin wires which exhibit plasma frequency in low frequency, say, GHz range will be introduced. Theoretical derivation and examples will give a physical vision to thin metallic wires. The point of “thin” wires is extremely crucial because thick wires would not reach the same function of negative permittivity.
In our knowledge, metals in the visible region and ultraviolet display a plasmon which is a collective oscillation of electron density. The charge on the electron gas is compensated by the background nuclear charge in the state of equilibrium. Under the effect of electromagnetic force, the negative electron gas and a surplus of uncompensated charge are generated at the ends of the specimen. This supply a restoring force between the opposite charges following a simple harmonic motion,
2
where e is the electron charge, d is the density of electrons, and m is the effective eff mass of electrons. With the interaction with electromagnetic radiation, the plasmon produces a dielectric function of the form,
2
where γ is the damping factor representing dissipation of the plasmon’s energy into the system. Meanwhile, it is small relative to ωp.
Figure 2.4: Dielectric function at lower frequency shows the domination of imaginary part. The real part is marked by clear circles while imaginary part is labeled by solid line.
Nevertheless, Eq.2.6 will become imaginary in the GHz frequency range because of the inevitable dissipation. As shown in Fig. 2.4, the imaginary part of permittivity is much lager than the real part. In other words, the dielectric function at lower frequency is dominated by the imaginary part, which is exactly out of our expectation.
Hence, a composite material that translates the characteristic feature of metallic response at ultraviolet region into GHz range is desirable
Figure 2.5: An array of infinite wires aligned with the z axis and arranged on a square lattice in the x-y plane. In the structure we considered, a might be a few millimeters and r a few microns [8].
In 1996, Perndry et al. [6] proposed a composite structure built by metallic thin wires which depressed the plasma frequency to GHz range. As shown in Fig. 2.5, thin metallic wires will confine the motion of electrons, then changing the effective mass of them. Because only part of the space is filled by metal, the average electron density is reduced to
Another factor we have to consider is an enhancement of the effective mass of the electrons caused by magnetic effects. This self-inductance gives an additional contribution to the momentum of eA, and therefore the new effective mass of electrons is given by
With the enormously enhanced effective mass and decreasing electron density, the plasma frequency will shift from ultraviolet region to GHz range, which is a correspondingly large amount. A sample of actual calculation will give below.
Example: aluminum wires
Form this mathematic work, a clear view of our composite metallic thin wires can be seen. The structure successfully decreases the plasmon frequency in a huge amount; hence the negative permittivity in microwave range is available. After solving the requirement of negative permittivity, the possibility of negative permeability will be discussed.
2.3 Ring structure of negative permeability
In the above section, the wire structure which conceptually replaces the atoms and molecules of a real material and presents negative permittivity property is introduced.
Although naturally occurring magnetic monopoles do not exit, a structure utilizing the concept of Drude-Lorentz model can bring negative permeability into reality. In 1999, Pendry et al. proposed a microstructure mimicking ordinary uniform material, and successfully showed the negative permeability [3].
(a) (b)
Figure 2.6: (a) Schematic layout of a single split ring resonator (SRR). A SRR has two conductive loops with a gap inserted. There is a small space between two loops, which can produce inner capacity. (b) The composite media of SRR array. The lattice constant a, which is about one tenth of the wavelength, is a crucial factor of this media [3].
As shown in Fig. 2.6, this structure consists of split ring resonator (SRR) array where r is the radius of the inner split ring of each SRR, c is width of each ring, and d is space between two split rings. Each SRR which behaves like an atom or a molecule has two conductive loops with a gap inserted. This gap prevents current from following around any one ring. However, there is a considerable capacitance between the two rings, which enable current to flow. The greater the capacitance between the two loop, the larger the current induced. The capacitance between different SRRs, moreover, must be considered in our calculation. Sequentially, the self-inductance of large capacity in and between the SRRs, which cause electrical
current following in two separating loops, will play the role of magnetic dipole. In this case, the permeability will satisfy
2
=π is the filling factor which represents the fractal area occupied by SRRS of each unit cell,
0
is the frequency where µeff diverges or resonant frequency. The symbol l is the distance between two layers, ρis the resistance of unit length of the sheets measured around the circumference, and c is the light velocity in vacuum. While a perfect 0 conductivity material is used, the term of dissipation will vanish as well; hence Eq.2.6 can be modified as
ωmp is the magnetic plasma frequency. The relation of effective permeability, ωmp, and ω0 is shown in Fig. 2.7. The region from ω0 to ωmp is what we desired for this micro-scale structure.
Figure 2.7: Typical curve showing the effective permeability of SRR array. Permeability
locating in the region between ω0 and ωmp exhibits expected negative value [3].
For SRR with parameter a = 10 mm, c = 1 mm, d = 0.1 mm, l = 2 mm, and r = 2 mm, the estimated resonant frequency will be
2 21
0
7.1 10
ω = ×
0
13.5 GHz,
mp14.4 GHz
f f
⇒ = =
Consequently, negative permeability can be realized in microwave range. The band desired in our calculation is pretty narrow, say, about 10%. In the forbidden band, a propagating plane wave in this composite SRR media will attenuate smoothly and eventually vanish because of the imaginary term in wave factor, say,
0 0
k = ⋅ =n k i ε µ ⋅k in ei k r(K K⋅ −ωt) . In other words, the transmission power of electromagnetic wave after passing through a SRR media is barely detectable. Thus, as shown in Fig. 2.8, a band-stop like characteristic in X-band spectrum is the classic curve of a SRR media.
Figure 2.8: Transmission spectrum showing the characteristic of a single SRR. The band-stop like behavior is due to the negative permeability caused by SRR structure [9].
2.4 Experimental Demonstration of LHMs
In the above two sections, the accomplishments of negative permittivity and negative permeability are described in detail. However, the demonstration of negative refraction index is still not available. Although the materials of both negative permittivity and negative
permeability are separating obtained, we are not sure if the combination of periodic wires and SRRs will work successfully. Will unpredicted interaction between wires and SRRs destroy the behavior of their own?
In 2000, Smith et al. presented the numerical simulation and experimental data to prove simultaneously negative permittivity and permeability effect. He fabricated the SRRs and wires on commercial available printed circuit board [9]. In his experiment, square arrays of SRRs and wires were constructed with a lattice spacing of 8 mm between elements. The result of transmission experiments on SRRs alone (solid curve), and SRRs with wires (dashed curve) are shown in Fig. 2.9.
Figure 2.9: Experimental result for LHMs. Solid line is the transmission spectrum of SRRs alone while dashed line is the transmission curve of SRRs with wires. A pass band of LHMs about 5 GHz is observed [9].
Following the conclusion that we have made in Sec.2.3, the transmission power will attenuate between f and 0 fmp as the solid curve shown. However, while SRRs are united with periodic wires, a negative refraction index will be obtained by simultaneous negative permittivity and permeability. The wave which originally attenuates will thus propagate due to a real wave factor (from i ε µ ⋅k0 to − ε µ ⋅k0). On the other hand, the wave which originally propagate (for instance, above 5.5 GHz) will thus become attenuating wave according to imaginary wave factor as well. Therefore, the transmission power locating on the absorption band previously exhibit a pass band characteristic basing on effect of negative
refraction index.
Although the demonstration of the composite LHMs has been completed, a more directly evidence of negative refraction index is still unavailable. We need a straightforward experiment result to convince everyone the existence of negative refraction index.
(a) (b)
Figure 2.10: (a) Photograph of the LHM sample. It consists of square copper SRRs and wire strips on glass circuit board. The rings and wires are on opposite sides of the boards. (b) Diagram of experimental setup. The black arrows represent the microwave beam as would be refracted by a positive index sample [4].
Figure 2.11: Experimental data of measuring refractive angle of LHM and Teflon.
The positive degree of Teflon provides the reference for LHM while the negative degree offers the evidence of negative refraction index [4].
In 2001, Smith et al. performed and experiment of Snell’s Law with a wedge composed of the LHMs (Fig. 2.10) [4]. By measuring the scattering angle after microwave beam transmitting the wedge, the effective refraction index can be determined. The refractive power peak of Telfon should be positive degree, as shown in Fig. 2.11, corresponding to positive refraction index. However, the power peak of microwave after passing through the LHM wedge locates at -61 degree, from which we deduce the index of refraction to be − ±27 0.1. Hence, the measurement successfully demonstrates the existence of negative refraction index at first time.
Chapter 3
Coplanar Structure of Metamaterials
The concept of negative permeability at certain frequency caused by SRRs is most interesting because the resonant frequency has strong relation with unit length and lattice constant. Recently, deformed SRR (DSRR)[10] [11] has provided a practical way to decrease the size of lattice constant, which is very useful while the short wavelength electromagnetic wave propagating in LHMs is desired.
Figure 3.1: A single DSRR with the property which can increase the resonant frequency with the same lattice constant as SRR.
When it comes to optical region, DSRRs whose scale is only about nanometers can be manufactured through the semiconductor process. However, there is still another problem companied by semiconductor process although it provides a practical method of such a small scale. Semiconductor process has a limitation that the metallic patterns can only be placed on a single plane, which means the wires and
DSRRs must be fabricated separately. In such a small scale, the orientation between wires and resonant rings will be a difficult problem and is hard to solve. Hence, we need a novel metallic structure, which consists of coplanar wires and resonant rings altogether in order to avoid those inconvenient factors in semiconductor process.
3.1 Design for Coplanar Structure of Metamaterials
As our knowledge, the construction of metamaterials has two parts. One is the ring structure which presents the negative permeability, and the other is the periodic wire which presents the negative permittivity. Hence, the design for coplanar structure must be divided into two parts – negative permeability and negative permittivity.
That is, we have to modify the original SRRs and wires; and combine them at the same plane. In the next two subsections, a symmetric DSRR will be introduced to give the possibility of coplanar structure. Because the lattice constant of such a symmetric DSRR is twice as the original DSRR, the wire structure with the same lattice constant may not exhibit negative permittivity at resonant frequency.
Therefore, discontinuous wires with different length must be considered in order to compensate the insufficiency.
3.1.1 Symmetric DSRR
In order to place the wire structure in the middle of ring structure, previous SRRs or DSRRs structures need to be modified. For SRRs, it is impossible to put metallic wires in the middle of them since they will cause inevitable contact with each other.
Such contact will finally bring unpredicted phenomenon that would destruct our desired efforts. On the contrary, DSRRs, which is distinct from SRRs, is possible to place metallic wires in the center of the rings. The gap between the two halves of each DSRR provides the possibility to build a coplanar structure. The most direct way to implement such a coplanar structure is to rotate DSRRs by 45 degree and redistributed the two parts of one DSRR equally.
Figure 3.2: Schematic diagram of a single symmetric DSRR where a = 1.531 mm, b = 0.655mm, c = 1.000 mm, and d = 4.988 mm.
As Fig. 3.2 shown, a modified DSRR called symmetric DSRR is presented. The gap c allows the existence of metallic wire without any contact; hence the functions of rings and wires can perform separately. Meanwhile, the total unit length of this symmetric DSRR increases from 2.62 mm to 4.988 mm, which indicates that the original lattice constant, 5mm, is not suitable anymore. A wider lattice constant is necessary for this symmetric DSRR. In our concern, default lattice constant is set to be 10 mm which is twice the length of original DSRR.
3.1.2 Variation of Wire Length
After solving the negative permeability structure by proposing a model of symmetric DSRRs in the above subsection, we have to continue our article on negative permittivity of periodic wires. As mentioned before, the plasma cutoff frequency is strong related to wire width and lattice constant. Here, by changing angular frequency to frequency and replacing Eq.2.8 and 2.9 into Eq.2.6 and 2.7, the cutoff plasma frequency and dielectric function will be
2 symmetric DSRRs, the lattice constant of periodic wires should also be 10 mm.
However, the cutoff plasma frequency will be far below desired operating frequency while possessing such lattice constant.
Figure 3.3: The relation of frequency and permittivity. Circle sign presents the real part of permittivity and plus sign presents the imaginary part of permittivity. The cutoff plasma frequency will locate at 5.84GHz while the lattice constant is 10 mm, the wire radius is 0.15 mm, and the conductivity of copper is 5.8×107 S/m.
A more mathematical example is given for further acknowledge. Here copper is chosen as the fabricating material and wire radius is set to 0.15 mm. The estimated cutoff plasma frequency will be about 5.84GHz by calculating through Eq.3.1 and 3.2.
In other words, the metallic wires will not behave like negative permittivity structure
In other words, the metallic wires will not behave like negative permittivity structure