Field distributions

Fig. 9(a) and 9(b) show the variation of the magnitude of the E-field components against the vertical y-direction for various frequencies within the first surface wave passband (0 to 10GHz according to Fig. 5(a)), whereas the graphs of Figs. 10. are for the H-field components.

As the frequency rises and moves deeper into the first surface-wave regime (2.05 through 9.05 GHz in 1GHz steps, as selected for plotting), the corresponding increased surface-wave phase constant k_z^univ beyond k_above  _{abv abv} and thus strengthened attenuation constant

above

y along the vertical y direction is indeed demonstrated by the progressively steepened exponential decay of the various field components with increasing frequency. In addition, the continuity of the |Ez|, |Hx| and |Hy| components across the y = d interface between the corrugations and the upper half region is observed as required.

Figure 9(a): |Ey| plotted against y-direction

23

Figure 9(b): |Ez| plotted against y-direction

Figure 10(a): |Hx| plotted against y-direction

24

Figure 10(b): |Hy| plotted against y-direction

Figure 10(c): |H_z| plotted against y-direction

25

III. Sectorial band gap

3-1 Dispersion diagram corresponding to Brillouin zone

For most of the studies in the literature, the EBG structures usually are used to deal with the entire bandgap, i.e. within the certain frequency bands, the surface waves are all suppressed on the EBG surfaces. There are potentially some applications which do not need a bandgap for all directions but just certain directions. Despite not potentially able to provide entire bandgap, planar corrugated surfaces are classically known to possess the capacity of offering surface-wave pass-bands and stop-bands along the directions parallel and perpendicular to the grooves and ridges, respectively, which are known as hard and soft surfaces as mentioned. However, no works have yet studied their candidature for serving as sectorial bandgap structures. We will demonstrate that planar corrugations are able to exude this capability. By capitalizing on the rapid surface-wave solution provided by the ACBC, we shall use the planar corrugated surface as the vehicle to illustrate how sectorial bandgap structures can be designed efficiently. This is something which no other periodic structures without analytic surface-wave solutions can readily afford.

Before we demonstrate the idea of sectorial band-gap and band-pass, we need to introduce the basis of the dispersion diagram and Brillouin zone. Brillouin zone is the set of

φ

k

_z

k

_x

Figure 11: Brillouin zone

26

wavenumbers which can describe the propagation of electromagnet ic waves in two-dimension photonic crystals, as shown in Fig.11, where k0 is the wavenumber in free space.Since we only consider the surface wave, the wavenumber which is vertical to the corrugation must be pure imaginary, which leads to

2 2

0

surface x z

k



k



k



k

(40)

so we just need to consider the region outside the circle for the surface wave case. As shown in Fig 11, each arrow constitutes a certain surface wavenumber for a certain frequency, but for the most important aspect here, there is not just only one surface-wave vector-arrow for any one certain frequency. This will be discussed deeper later.

In Section II, we show four parts (O→Z, Z→M, M→X, and X→O) of the dispersion diagram. As mentioned, any one of these parts is obtained by fixing two of these three unknowns, (i) kx, (ii) kz and (iii) frequency, and then the roots for the remaining unknown are solved for. If we set kx as an unknown, and set

tan( )

x z

k  k 

(41) where φ is the angle between the propagation path of the surface wave and the z axis. Then

Figure 12(a): Dispersion diagram for oblique wave as φ is 15 degrees

27

the roots for kz are solved we can get the dispersion diagram for the surface waves propagating along the path which has φ degrees with the z-axis. For the extreme case, we can take the X→O part as the 90 degrees cases turning from O→Z part. Two arbitrary different examples are shown in Fig. 12, the dimensions are as follows: dz = 3mm, d=4mm, εrel

,

groove = 3 and the values of φ are 10 and 45 degrees, respectively.

3-2 Concept of sectorial bandgap

Refer to the fictitious dispersion diagram in Fig. 13 below, which shows traces for various azimuth  (measured from the z-axis perpendicular to the corrugations) directions of surface-wave modal propagation, i.e. each trace pertaining to a certain fixed , with the original dispersion paths of OZ, Zbeyond included for reference. Let the frequency corresponding to the Brillouin limit be denoted as f₁. The associated surface-wavevector at this frequency is shown by the arrow in Fig. 14 with magnitude ksurf f1

= /dz and directed along z perpendicular to the corrugations. As the surface-wavevector enters the oblique nonzero  regime ( measured from the z-axis perpendicular to the corrugations), but with kz

maintained at /dz, i.e. now the surface-wavevector component along z (kz) is no longer zero, Figure 12(b): Dispersion diagram for oblique wave as φ is 45 degrees

28

both the eigen-frequency and modal surface-wavenumber increases further, the relationship between them as indicated by the traces corresponding to k_surf values greater than k_surf ^f1 = /dz. For illustration, the surface wavevector at an example frequency of f2 (labeled in Fig. 13) is represented in Fig. 14 by the arrow with magnitude ksurf f2

. Due to symmetry about both horizontal and vertical axes, two arrows with magnitude ksurf f2

are shown in Fig. 14 (the same applies for other oblique surface-wavevectors). As the frequency rises further up to the point where the next higher-order surface-wave mode starts to emerge at f3 as shown in Fig. 13, the surface-wavevector is directed towards an even larger  angle as represented by the arrow with magnitude k_surf ^f3 in Fig. 14.

As it can be seen in Fig.13, the original stopband region for zero angular span (phi=0) is between f1 and f3, and for the case when phi is “φb” degrees, the stopband zone is from f2 to f3, so it means as the phi gets larger, the stopband areas get smaller. In other words, we can say that during the period from f2 to f3, there is “at least” “φb” degrees sectorial band gap area. It is easily misunderstanding the above idea in another expression which is Brillouin zone as shown in Fig.14. The sectorial bandgap angle (SBGA, in degrees) is between f1 and f2, but not between f2 and f3. Also by applying the above idea, we can define the boundary between the conventional soft and hard surfaces, which is the surface-wavevector ksurf f3

in Fig.14, i.e. the sectorial bandgap angle corresponding to the frequency which the next mode just appears.

Frequency

29

As mentioned, between frequencies f2 and f3, no surface wave can propagate within the sector between the two symmetric arrows with k^f2surf (each being the surface-wavevector at

f

2). We call the range between the f2 and f3 as sectorial bandgap width (SBGW=f3-f2, indicated in Fig.14), which means in this area, no surface-waves can propagate within the angular size with at least “SBGA (which is φb in this case)” degrees. But why is the SBGW upper-limited by f3? As the frequency just exceeds this frequency, the next higher-order surface-wave mode starts to appear and propagate along the x-direction. This of course falls inside the sector and thus disqualifies frequencies above f3 from being included in the SBGW.

3-3 Sectorial bandgap corrugations design

The theory of the sectorial bandgap had been introduced in the previous section, and it will be convenient if the relationship between the dimension of the corrugation and the sectorial bandgap angle is known. For example, if the height of the corrugation is known, and also the target frequency is given, it is possible to provide relationship between the groove material and the sectorial bandgap angle it can reach. The idea can be done by using the Eq. (31) again.

Substituting Eq. (31) by Eq. (41), setting the kz as Brillouin limit and an arbitrary frequency, so the relative permittivity of the groove can be solved as the roots.

Figure 14: Top view schematic of corrugation

Sectorial bandgap angle (SBGA, in degrees)

Sectorial bandgap width (SBGW, in Hz)

30

and g=1.6mm. Figure 15 shows O→Z path of dispersion diagram obtained by corrected ACBC method as the relative permittivity is 4.2, and as it is shown, the first and second stopbands start (which also means the initial point of the Z→M path) from 9GHz and 27.5GHz, respectively. Figures 16 show the relationship between the groove material and the sectorial bandgap angle, and three different frequencies are discussed here, which are 9GHz, 15GHz, and 27.5GHz. As shown in Fig. 16(a), there are two exponential curves which represent the first and second mode. Just as mentioned, 9GHz is the start of the first stopband, and it is shown that the first point (solution) of the first mode starts when the relative permittivity is 4.2, which corresponds to the initial condition. For the case as the frequency changes to 15GHz in Fig. 16(b), it is shown that both curves (modes) will move down, which means that the sectorial bandgap angle gets larger compared to Fig 16.(a) for the first mode.

As the frequency becomes 27.5 GHz, which is the start point of the second stopband, there are two roots when the relative permittivity is 4.2, as shown in Fig. 16(c). One thing should be mentioned here, once the new mode enters the passband, even though there has solution for the first mode, the bandgap will be covered by the second mode, so that the phenomenon will not be seen. There is one thing should be

Figure 15: OZ path of the dispersion diagram for present case

31

Figure 16(b): Relationship between relative permittivity and sectorial bandgap angle in 15GHz.

Figure 16(a): Relationship between relative permittivity and sectorial bandgap angle in 9GHz.

32

Figure 16(c): Relationship between relative permittivity and sectorial bandgap angle in 27.5GHz.

Corrugation

Normal periodic structures

Figure 17: Comparison between corrugations and normal periodic structures Surface wavenumbers

33

noticed that the predicted sectorial bandgap angle shown in Figs. 16 are ideal, which is because the loss tangent of the material in the grooves does not considered in ACBC method, so the results might have deviations, but the phenomenon still exist.

The same phenomenon may occur in other periodic structures, but the corrugation can achieve the best result. The reason is that we can notice that compared to the normal periodic structures, the slope of the Z→M part of the dispersion diagram for the corrugation is steep, as shown in Fig. 17. Since the slope is steep, the corrugation does not have “any direction”

bandgap, but this special characteristics caused the corrugation achieve the sectorial bandgap idea. For the normal periodic structures whose slopes are flat, the SBGW may be too small to clarify which cause the difficulty to reach the idea.

3-4 Simulation results

For the reality, it is difficult to get the dispersion diagram directly, and for most of the literature, the simplest way to explain the dispersion diagram will be the scattering parameters.

In this section, the simulation results for the sectorial bandgap will be shown, and we use the transmission coefficient to verify the sectorial bandgap theory. The results are verified by CST software.

Figure 18(a): O→Z part of the dispersion diagram as the material of the groove is FR4.

34

In this section, the dimensions we choose are as following: d = 4mm, dz=2.6mm, and g=1.6mm, also we examine two different cases, for which the material we choose are FR4 and Rogers RT6010, whose relative permittivity is 4.2 and 10.2 respectively. At first, the O→Z part of the dispersion diagram obtained by the ACBC methods for above mentioned dimensions are shown in Figs. 18. The start frequencies of the stopband in Fig. 18(a) are 8.8GHz and 26GHz, and about 17GHz and 29GHz in Fig. 18(b). Figs. 19 show the top view of the corrugation in CST simulations, and we use the two identical waveguides as sources and receivers (the cutoff frequency for the waveguides is 1.53GHz), since the limitation in CST setting, for verifying the oblique waves on corrugation surface, we rotate the corrugation for 30 degrees meanwhile fixing the distance of the waveguides. The transmission coefficients for two different cases are shown in Figs. 20, which represent the relative permittivity as 4.2 and 10.2, respectively. By observing the 0 degree in Fig 20(a), it can be seen that stopband bandwidth is wider than the prediction as shown in Fig 20(a). The reason is just as mentioned that it is because that in ACBC method, loss tangent of the material in the grooves is not considered, so that when dealing with the practical cases, some surface waves may decay in the grooves. For the material such as Rogers RT6010, the value of loss tangent is much smaller than FR4, so the simulation error compared to ACBC methods is also smaller, which is shown in Fig. 20(b). In Figs. 20, the transmission coefficients of oblique waves for 30

Figure 18(b): O→Z part of the dispersion diagram as the material of the groove is Rogers RT6100.

35

(a) Waveguide (b)

Figure 19: Top view of the corrugation in (a) 0 and (b) 30 degrees rotation in CST simulations.

Figure 20 (a): Transmission coefficient results using FR4 as the material of the grooves.

36

degrees are also simulated. Just as presumed, the stopband bandwidth gets smaller as the angle decreases. Take Fig 20(a) as example, 7.5GHz is f1 as the symbol in Fig. 14, 9 GHz is f2, and 12GHz is f3. So during 9GHz to 12GHz, we can get “at least” 30 degrees sectorial bandgap angle, and the SBGW is 3GHz (which is due to 12GHz-9GHz).

3-5 Measurement results

In this section, the measurement results will be shown. The dimensions of the corrugation we fabricate are as followings: dz = 2mm, g=1.6mm, d = 4.5mm, and εrel

,

groove = 4.3. The structure is shown in Figs. 21, and the dispersion diagram retrieved from ACBC method is in Fig.22. Two k-band (from 18GHz to 26.5GHz) horn antennas are used as the source and receiver, and they are put on the corrugation surface. Since the aperture of the horn antenna is tall from the surface, the absorbers are stuck on the apertures and just leave a thin gap in the bottom in order to be closer as a surface wave. In the beginning, two horn antennas are placed face to face, and theoretically, the energy is 100% through so that the value of S21 should be 0dB. But we want the wave to be more like a surface wave propagating on corrugation surface, the absorbers are added to block most areas of the aperture, so the energy is absorbed. The setting framework described above is shown in Fig. 23, and the section view is in Fig.24.As we can see in Fig. 25 that the reference is about -22dB. Then we put the corrugation under the

Figure 20 (b): Transmission coefficient results using Rogers RT6010 as the material of the grooves.

37

Figure 21(a): Top view of the fabricated corrugation

Figure 21(b): Side view of the fabricated corrugation

38

Figure 22: O→Z part of the dispersion diagram

Figure 23: Measurement framework.

39

Figure 25: Measurement results for the existence of the corrugation.

Absorber

Horn aperture

Figure 24: Section view

40

horns, and it can be observed that there exists a stopband region between 21GHz to 26GHz, for which this region is also the stopband area in dispersion diagram obtained from ACBC method as shown in Fig. 22. Figure 26 shows the comparison as the corrugation rotates 0 degree and 30 degrees. The trends are similar to the simulation results. The larger angles the corrugation rotates, the start frequency of the stopband will be higher, which means that the stopband bandwidth is smaller.

Figure 26: Comparison for the results as the corrugation rotates 0 degree and 30 degrees.

41

IV. Relationship between scattering parameters and dispersion diagram

4-1 Introduction

Most of the time, in order to obtain knowledge of just the width and location in the frequency spectrum of bandgaps of periodic structures, the use of two-port scattering parameters is more direct than the dispersion diagram. Nonetheless both approaches have their own benefits. Scattering parameters can characterize the center frequency, bandwidth, and the attenuation level of the stop band, while dispersion diagram can show the relationship between the frequency and wavenumber, further allowing the retrieval of data such as the phase velocity. For most research in the literature, dispersion diagrams are generated by full wave simulators, and when it comes to the measurement stage, the diagrams are difficult to gauge directly, so the scattering parameters are used to explain the dispersion diagram instead.

But scattering parameters are after all only intermediate results, shedding no insights at all aboupt the wavenumber at that frequency. It will be more convenient if we could know the relationship between the scattering parameters and dispersion diagram, and then we could use the directly measurable S-parameters to generate the measured dispersion diagram.

For a normal periodic structure whose equivalent circuit is not complicated, it is easy to get the expression of the wavenumber by ABCD matrix, which will be briefly introduced later.

But for a more complicated periodic structure, the equivalent circuit is complex, resulting in the difficulty to deal with the matrix in the previous method. Since the equivalent circuit is too complex to analyze, it is a good way to treat this complex unit cell as a uniform material. In other words, the scattering parameters are obtained at first, and then we reconsider the unit cell as a uniform material, deriving the relationship between the scattering parameters and wavenumbers, finally substituting the measured scattering parameters into the relationship. In this section, we will show the process of deriving based on a simple theory, and the accuracy for the transformed dispersion diagram is good.

In [16], Pozar analyzed a periodically loaded transmission line, and its equivalent circuit is shown in Fig.27, where d is the distance between two unit cells, and b is the value of susceptance. Since it can be seen as a cascade of identical two port networks, we can use ABCD matrix to relate the voltages and currents on either side of the unit cell:

1

42 Now consider the phase difference between the nth and (n+1)th terminals

1

exp( )

Substituting Eq. (45.a) and Eq. (45.b) into Eq. (42), we can get

1

For nontrivial solutions, the determinant of the matrix in Eq. (46) must vanish, leading to

cosh 2

d A D



  (47)

For a lossless periodic structure and symmetric network

c o s d   A

(48)

Figure 27: Equivalent circuit model of an infinite long periodically loaded transmission line.

jb jb jb jb jb

d

43

Since we know the relationship between β and matrix element A in Eq. (48), and also the relationship between matrix element A and scattering parameters in two-port networks is already known, we can easily get the dispersion diagram if we have scattering parameters data.

4-2 Theory

The relationship between the scattering parameters and wavenumbers introduced by Pozar is only suitable for simple cases. For a more complex periodic structure, the equivalent circuit is not easy to retrieve. Even though the equivalent circuit can be retrieved, the susceptance would be too complicated, causing difficulties with the calculation using the abovementioned method. To solve this problem, we can think of it from another perspective, which is by treating the complex unit cell as a uniform material whose effective relative permittivity and permeability are εr and μr, respectively, as shown in Fig.28. The representations of the reflection and transmission coefficients are stated as follow:

11 coefficient in Eq. (49) can be represent as

2 Utilizing the formula of sum of the geometric series, which is

0 Now we assume that both medium in the left-hand side and right-hand side of the material are air, resulting in

44

45 Substituting Eq. 14(a) to Eq. 14(d) into Eq. (13), we can obtain

12

Similarly, we can obtain the total transmission coefficient

21(1 23) exp( ) 21(1 23) exp( 3 )( 12 23) ...

Again substituting Eq. 54(a) to Eq. 54(d) into Eq. (57), resulting in

2

Eliminating the exponential terms in Eq. (55) and Eq. (58), we can get

2 2 2

12

S

11 12(

S

11

S

21 1)

S

11 0

       (59)

Also we can present Γ12 in terms of the intrinsic impedance:

2 1

Substituting Eq. (60) into Eq. (59) results in

2 2

46 exp(-jβl)=z, we can present refractive index as:

" ' structures, it is subject to the criterion dictating proper choices of the signs of η and n, which

" ' structures, it is subject to the criterion dictating proper choices of the signs of η and n, which

在文檔中 利用漸近邊界條件分析表面波於皺褶結構之色散 (頁 30-0)