3 Coplanar Structure of Metamaterials 18
3.4 Discussion and Conclusion
After examining all experiment and simulation results, the physical meanings of symmetric DSRRs and discontinuous wires need to explain. Our discussion will be emphasized on three aspects; they are the physical activity of symmetric DSRRs, response of wire structures, and the corresponding behavior between wires and symmetric DSRRs.
The symmetric DSRRs not only present the property of negative permeability but also display the main factor determining the resonant frequency. As mentioned previously, the experiment and simulation data of symmetric DSRRs exhibit the exact existence of negative permeability. The resonant happens as the propagating direction is parallel to the PCB, where magnetic field penetrates symmetric DSRRs.
At this moment, the external magnetic field passing the ring structure generates strong currents flowing along two half rings of a symmetric DSRR. Charges excited by this surface current would accumulate at the opposite proximity of each half part.
The positive and negative charges which occupy against the small gap hence introduce a strong electric field. The scheme of symmetric DSRRs is quite similar with that of DSRRs; thus it is not surprise to see symmetric DSRRs having the resonant mechanism. Furthermore, the resonant frequency of symmetric DSRRs which locates at 14.41GHz is another interesting issue to discover. The experiment results of symmetric DSRRs introduce extra accounts, the total cell unit number, involving the determination of resonant frequency. Resonant frequency should be dominated by lattice constant due to anterior knowledge. Nevertheless, the experiment data show that the resonant frequency of symmetric DSRRs with 10mm lattice constant is higher than the that of DSRRs with 5mm lattice constant. Former sample has a resonant frequency at 14.41GHz while latter possesses around 12GHz [10]. The experiment results can barely find a reasonable explanation if the results are only investigated by viewing the lattice constant. Consequently, another concern such
as the total number of units on a single sample should be included as the effective factors. Comparing to the sample of DSRRs [10] whose total number of units is 560, the total number of symmetric DSRRs in this thesis is sole 100. As we know, the resonant frequency has mighty relation with the unit number. A sample with identical lattice constant would have a lower resonant frequency when it holds more unit numbers. Therefore, with only one forth of the total unit number, symmetric DSRRs display higher resonant frequency even having twice the lattice constant. In summary, the experiment results of symmetric DSRRs are not surprised since the phenomenon of negative permeability is observed on the fundamental of the relationship between resonant frequency, lattice constant, and cell number.
Next, we want to explore the behavior of wire structures. As the experiment and simulation data shown, the response of wire1 to wire3 is barely distinct, which makes the physical explanation of them being similar. Generally, the cut wires of wire1 to wire3 only have reaction to the external incident electric field at 90 degree. In other words, the effect of negative permittivity only appears when the microwave encounters periodic wires along the propagating direction. Also, the absorption area of discontinuous wires does not extend to zero frequency on account of the intrinsic traits. In contrast to the behavior of wire1 to wire3, long cut wire, wire4, carries much disparate activity which is not actually representative negative permittivity structure.
The experiment data indicates that it has stronger absorption at 0 degree rather than that at 90 degree, which means wire4 possesses different characters other than negative permittivity. The phenomenon that absorption intensity decreases as the rotation angle increases from 0 to 90 degree reminds us the physic reaction of grating.
When the plate of PCB is perpendicular to the propagating direction, each metallic strip of wire4 on PCB just mimics the line grating in ordinary optical experiment.
Hence the chance of constructing metamaterials will tends to zero even we put wire4 and symmetric DSRRs together. For accomplishing the goal of building negative permittivity structure, wire1, wire2, and wire3 have demonstrated the existence of it except for wire4.
After discussing ring and wires respectively, the reaction for the medium comprises these two structures needs to be discovered. The response of composite
symmetric DSRRs and wires approach the behavior of symmetric DSRRs alone according to Fig. 3.8, 3.9, and 3.10, which means their interaction with the peripheral electromagnetic waves is primarily dominated by the characteristics of symmetric DSRRs. On the contrary, wire4 is the major trait that controls the spectrum of composite wire4 and symmetric DSRRs. In Fig. 3.11, the bold line curve indicates the original response of symmetric DSRRs is totally destroyed by an introduced interruption, say, wire4, within the ring gap. The apparent absorption band which extends from 12GHz to 16GHz caused by symmetric DSRRs is eliminated, and suggests that the ring structure of composite wire4 medium does not have function as expected; external electromagnetic waves may ignore such a structure as propagating.
In fact, the intuitive physical sense tells the domination of long cut wires in long wire compound and rings in short wire compound. Wire length between 2.135mm and 3.062mm should have a crucial point which may possess the property of negative permeability and negative permittivity simultaneously in our prior concept. However, the experiment data is against such a deduction. Fig. 3.9 and Fig. 3.10 point out the composite medium still holds the absorption curve even when short wires are inserted.
The truth is that the composite medium will always has the absorption activity no matter how long the wire is, and the absorption is caused by either wire or symmetric DSRRs. While the wire is short enough, the behavior is controlled by symmetric DSRRs. Once the wire exceeds the length, wire structure will be takeover the response immediately. Therefore, the scheme of symmetric DSRRs and discontinuous wires is not a successful mechanism realizing metamaterials eventually.
Chapter 4
Double Resonant Frequency (DRF) DSRR
Since the concept of metamaterials had been proposed in 90s, it has inspired a lot of attention and myriad ideas of application. The ultimate target of developing metamaterials is to apply such a novel structure at the range of visible light so that people can break conventional use on optical systems nowadays. That is why many works accomplished focuses on reducing the size of metamaterials. However, even the difficulties of manufacturing issue for metamaterials at light region are concurred, there is still a serious problem involving terrible power efficiency for realistic applications. In this chapter, the idea of fractal-like will be utilized to compensate the power loss. First, the original concept of fractal and the inheriting fractal-like idea for double resonant frequency (DRF) DSRR will be introduced. Then two types of fractal-like structure are proposed as possible solutions for this problem. Finally, physical explanations and discussions will be presented.
4.1 Inspiration of Fractal-like
4.1.1 Concept of Fractal
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in straight line,” the first few sentences described in the book written by Mandelbrot in 1977 was the most central concept of fractal [13]. Before the
concept of fractal had been proposed, the world had been dominated by Euclid’s “Elements”
which clearly defined the integer dimension of all objects on earth. Sets and functions that are not regular or smooth enough will be ignored and not worthy for study. The attitude, however, started to change step by step about 100 years ago. Some mathematicians proposed a few
“monsters” figures showing untraditional geometric configurations. They also defined new dimension which is not being integer for these monsters. The study for such untraditional geometric frames continued in the following decades. In 1977, due to the development of computer, Mandelbrot summarized all ideas and named “fractal” to this field of study. He claimed that many patterns of nature are so irregular and fragmented that nature exhibits not simply higher degree but altogether different level of complexity. Therefore, he conceived and developed a new geometry of nature which can describe many practical cases in the world.
Also, the most important role in this work is the fractal dimension (Hausdorff dimension) that describes configurations in a way different from Euclid’s concept.
Figure 4.1: Construction of the middle third Cantor set F, by repeated removal the third of intervals [14].
The middle third Cantor set is one of the best known and most easily constructed fractal; nevertheless it displays many typical fractal characteristics. The construction of it will begin from a unit interval by a sequence of deletion operations as shown in Fig. 4.1. The first step is to delete middle third of L so that only interval [0,1/3] and [2/3,1] survive. Similar recursive steps will continue from L1 to L2 and so on. Here,
L
k consists of 2k intervals each of length 3−k and F may be thought of as the limit of the sequence of setL
k as k tends to infinite. It is obviously impossible to draw such an infinitesimal set of F hence picture of F tends to be one of theL
k, which is a good approximate of F when k is reasonably large. In fact, the set of F will be seen as infinite points (which are considered as zero dimension in traditional geometry) instead of segments of a single line (which are considered as one dimension in traditional geometry) while k tends to infinite. The intuitive instinct is slight different from geometric definition thus the fractal dimension, which is first proposed by Hausdorff, is necessary. The definition of fractal dimension is :ln( ) ln(1/ ) D Nδ
=
δ
(4.1)where Nδ is the number of segment after each iterative step and
δ
is the ratio of the length after each operation to that before each operation. From Fig. 4.1, the fractal dimension of middle third Cantor set will beln( ) ln(2)
0.631 ln(1/ ) ln(3)
D Nδ
=
δ
= = (4.2)Figure 4.2: Construction of the von Koch curve, by repeated removal the middle third of each ling segment and replacing it by another two segments equal to remain parts [14].
The fractal dimension of the middle third Cantor is neither one dimension (1-D) nor zero dimension (0-D). It is a kind of configuration locating between 0-D and 1-D. A
further example of famous von Koch curve shown in Fig. 4.2 can give an even clearer vision of fractal dimension. The initial length of Koch curve is set to be unit length. L1 consists of four segments obtained by removing the middle third of L and replacing it by the other two sides of the equilateral triangle based on the removed segment. In other words, the parameters Nδ and
δ
are 4 and 1/3 respectively thus the fractal dimension will be 1.262 which is neither 1-D nor 2-D structure. In fact, as k tends to infinite in developing von Koch curve, the set F will approach filling the whole surface of the trace which seems to be 2-D frame while it will be still considered as 1-D frame in traditional geometry.In conclusion, the concept of fractal could be used to describe most of the patterns which are originally thought to be irregular or amorphous existing in our mother nature. Since Mandelbrot proposed this concept in 1977, it has been widely applied to various areas including biology, geology, astronomy, and chemistry. People in diverse fields try to find out a rule explaining for most frames through fractal. With no exception, there are also numerous studies of fractal for electromagnetism. The best known application of fractal for microwave is the fractal antenna which can promote the bandwidth and radiation pattern due to its high characteristic of self-similar. Therefore, basing on the successful experience of fractal antenna, we wonder if it is possible to apply fractal or fractal-like techniques on metamaterials.
Could it help to countervail the insufficient power transmission of intrinsic metamaterials just like it does for the fractal antenna? In the next subsection, the combination of fractal-like skill and metamaterials will be introduced in order to discover the possibility of compensating the power loss of metamaterials.
4.1.2 Implement of Fractal-like Concept on Metamaterials
Power insufficiency is always a problem of metamaterials when it is utilized within the range of visible or infrared range. Fortunately, the successful experience of fractal antenna brings some hints for conquering the unsolved issue. Basing on the extension of the percept of fractal, a DSRR structure contains two different lattice constants mimicking fractal is proposed and shown in Fig. 4.3.
(a) (b)
Figure 4.3: Two different implementations of double resonant frequency DSRR. (a) One of the methods to construct the structure with two lattice constants at the same time. (b) Another way to build up double resonant frequency DSRR.
These two patterns are not actually fractal because they do not obey its strict definitions; they are just similar to the concept of fractal, so that is why they are called fractal-like. Since this structure has large and small DSRRs simultaneously, there should be double resonant frequency while doing the experiment. In the design, there are two methods to realize the idea of double resonant frequency. The first one is to fill up the larger DSRR by smaller ones, and another one is to excavate smaller DSRRs from a complete larger DSRR. We hope that there would be some interaction between two kinds of DSRRs in different scales while their corresponding resonant frequency is observed respectively. Moreover, if there is indeed some interaction, the absorption power of them could be utilized to retrieve the insufficient power of the other one. In the next two sections, details of implementing these two double resonant frequency samples will be introduced and described clearly. Some experiment results and discussions will also be included as the demonstration.
4.2 DRF-DSRR Basing on Split DSRR
Divide the larger DSRR into numerous small DSRRs is the first method to realize the double resonant frequency structure, thus this structure is named Split DSRR
(SDSRR). While the small DSRRs is small enough, the lager one will be considered as a complete unit of negative permeability medium. This novel property may provide the solution for the unsolved power issue. In this section, the implementation of DRF-DSRR basing on SDSRR will start from a pre-experiment with a small gap on the ordinary DSRRs.
4.2.1 Pre-experiment:
Design and Experiment of SDSRR
Before start the experiment of double resonant frequency basing on SDSRR, a pre-experiment needs to be done first in order to obtain preliminary evidence to demonstrate feasibility of SDSRR. Moreover, the experiment results can provide useful information for further estimation as well.
Design
The idea of pre-experiment is to dig several gaps in the middle of DSRR just like Fig.
4.4 shown. The schematic drawing of SDSRR is similar to DSRR except for four straight gaps whose width is 0.2mm. Despite the gap, other restrictions such as linewidth d and lattice constant are all the same with that of original DSRRs.
Figure 4.4: Schematic figure of a single split DSRR (SDSRR) shows four straight gaps g whose width is 0.2mm. Linewidth d is 0.655mm and w is equal to 0.555mm.
Imitating a completely original DSRR by so many small DSRRs is the initial
inspiration of accomplishing DRF pattern. In such a structure, gaps between different small DSRRs are the most crucial factor of success. Therefore, the design of SDSRR which holds four gaps is set to be the pre-experiment, and helps to supply proper knowledge. If the absorption will appear even with gaps within DSRRs, then there is great chance for SDSRR to build up the DRF pattern. On the contrary, if the reaction between SDSRR and external microwaves does not present apparent absorption, the possibility of implementing SDSRR into DRF pattern will be quite minute.
Sample Specification and Experiment Environment
In experiment, SDSRRs alone will be measured. The cooperative response of them with continuous wires whose radius is 0.15mm and plasma frequency is 18.20GHz basing on Sarychev and Salaev’s deduction [15] will be detected as well. The lattice constant of SDSRRs is still 5mm which is only half of that in Ch.3. Here SDSRR patterns are made by copper. All metallic media that consist of periodical arrangement of wires and SDSRRs are fabricated on PCB, whose specification is the same as that in Ch.3. However, in this experiment, metallic wires and SDSRRs are manufactured in opposite faces of a single PCB, thus wires and SDSRRs will not disturb each other. The total number of units on per PCB is 500 (25 cells in one row and 20 cells in one column). Moreover, the experiment environment is exactly the identical to that in Sec.3.2.1.
Experiment
The spectrums of SDSRRs and SDSRRs with wires at 90 degree incident are displayed in Fig. 4.5. Absorption curve of SDSRRs themselves only appear while the rotation angle approaches 90 degree; absorption phenomenon does not occurs as the incident microwave is perpendicular to the patterns. Thus, the discussion can focus on the behavior of SDSRRs at parallel incident condition. First, free space reference marked by dot line presents the intrinsic spectrum of network analyzer itself. The variation of free space response is acceptable since the relative response between free space and SDSRRs is more important than the absolute values of absorption. Next, we are going to discuss the behavior of SDSRRs themselves. The maximum drop of
SDSRRs happens at 14.94GHz, and it has a difference of 29.49dB corresponding to free space reference. This power drop presents the existence of negative permeability even when there are four gaps across the linewidth of each original DSRR. When the periodic wires are adding to the opposite plane against SDSRRs on a single PCB, the cooperative behavior is presented in Fig. 4.5 as well. A manifest enhancement that approaches 26dB around 15GHz is observed, which means successful experiment results as expected. In other words, real propagation constant with negative value is obtained when metallic thin wires possessing negative permittivity and SDSRRs holding negative permeability both exhibit simultaneously around 15GHz.
Figure 4.5: Transmission properties of SDSRRs and their response adding wire structure at parallel incident. The free space reference is presented by dot line; the result of SDSRRs themselves is shown by clear circles, and the response of composite wires and SDSRRs is denoted by bold solid line.
The success of SDSRRs is extremely important because it maintains the fundamental properties of negative permeability even being modified by excavating several gaps.
This evidence could provide truly preliminary information and support that there would be great chance for SDSRRs to accomplish DRF pattern, thus it is worthy doing forward design and experiment in the next few subsections.
4.2.2 Design for DRF-DSRR Basing on SDSRR
Figure 4.6: Schematic layout for DRF-DSRR basing on SDSRR. Each lager DSRR consists of 40 small DSRRs. The lattice constant a is set to be 60mm while the lattice constant of small ones is 4mm. The linewidth of small DSRRs is still 0.655mm.
The overall design of DRF-DSRR basing on SDSRR is presented in Fig. 4.6 as the
The overall design of DRF-DSRR basing on SDSRR is presented in Fig. 4.6 as the