2.1.1 Circular waveguide with dielectric lining on metal wall

This model (model-1) includes a metallic circular waveguide of inner radius rW, inner wall of which is containing a dielectric lining of inner radius rL and relative permittivity ε<sup>r</sup> for the full length of the waveguide [7]. (Here, the waveguide is considered to be infinitely long and there is no reflection of the traveling signals

Figure 2. Circular waveguide with dielectric lining on metal wall [7].

Figure 3. Circular waveguide with dielectric coaxial insert [7].

frequencies above this lower cutoff frequency are allowed to propagate through the waveguide, and the signals having frequencies below this frequency will face a high reflection. Because of this characteristics a waveguide is inherently a high pass filter. The waveguide supports two kinds of velocities namely the phase velocity and the group velocity. The phase velocity at a chosen frequency is the one with which the signal of constant phase travels, which is represented by the slope of a line joining a chosen frequency point on ω � β dispersion characteristics to the origin (point representing zero frequency and zero phase propagation constant), i.e. mathematically given as ω=β. The group velocity at a chosen frequency point is the one with which the energy in signal travels, which is represented by the slope of the ω � β dispersion characteristics at the chosen frequency point, i.e. mathematically given as dω=dβ (Figure 1). Thus, one can control the supported phase and group velocities in a waveguide by tailoring its dispersion characteristics. Such tailoring can be achieved by loading the waveguide by metal and/ or dielectric in to the smooth wall waveguide [1–5]. The characteristics (propagation or dispersion) of the conventional (smooth wall) waveguide changes with the metal and/or dielectric loading, and the same cannot be generated naturally. Therefore, the metal- and/or dielectricloaded waveguide may be considered as artificially created material or artificial material. In part of the chapter to follow, a number of circular waveguide models containing various metal and/or dielectric loading are considered (Section 2). The electromagnetic boundary conditions (Section 3) and the dispersion relations (Section 4) of these loaded waveguides are outlined. Further, the dispersion characteristics of all the considered loaded waveguides are discussed with their sensitivity against variation in structure (geometrical) parameters (Section 5). Finally, the

ω � β dispersion characteristics of a circular waveguide showing the waveguide cutoff frequency and phase and

Although, the considered structures being a single conductor structure support TE (Ez ¼ 0) as well as TM (Hz ¼ 0) modes, they are being analyzed for the TE modes. The structures excited in these modes are of the interest for a specific class of vacuum electronic fast-wave devices, specifically the gyro-devices. In the

conclusion is drawn (Section 6).

2. Structure models

274

Figure 1.

group velocities.

Electromagnetic Materials and Devices

from the waveguide extremes). Thus the radial thickness of the dielectric lining can be calculated as rW � rL. For the sake of analysis, the structure may be divided into two analytical regions, the central free-space (dielectric free) region I: 0≤r , rL, 0≤z , ∞; and the dielectric filled region II: rL ≤r , rW, 0≤z , ∞ (Figure 2). The relevant (axial magnetic and azimuthal electric) field intensity components may be written as [7]:

In region I:

$$H\_x^I = \sum\_{n = -\infty}^{+\infty} A\_n^I J\_0\left\{\boldsymbol{\gamma}\_n^I \boldsymbol{r}\right\} \exp j(\boldsymbol{\alpha}\boldsymbol{t} - \boldsymbol{\beta}\_n \boldsymbol{\varepsilon}) \tag{1}$$

$$E\_{\theta}^{I} = j\alpha\mu\_{0} \sum\_{n=-\infty}^{+\infty} \frac{1}{\mathcal{I}\_{n}^{I}} A\_{n}^{I} J\_{0}^{\prime} \{\boldsymbol{\chi}\_{n}^{I} \boldsymbol{r}\} \exp\boldsymbol{j}(\boldsymbol{\alpha}\boldsymbol{t} - \boldsymbol{\beta}\_{n} \boldsymbol{z})\tag{2}$$

In region II:

$$H\_x^{\Pi} = \sum\_{n = -\infty}^{+\infty} \left( A\_n^{\Pi} J\_0 \{ \chi\_n^{\Pi} r \} + B\_n^{\Pi} Y\_0 \{ \chi\_n^{\Pi} r \} \right) \exp j(\omega t - \beta\_n \mathbf{z}) \tag{3}$$

$$E\_{\theta}^{\text{II}} = j\boldsymbol{o}\,\mu\_{0} \sum\_{n=-\infty}^{+\infty} \frac{1}{\mathcal{I}\_{n}^{\text{II}}} \left( A\_{n}^{\text{II}} J\_{0}^{\prime} \{ \boldsymbol{\chi}\_{n}^{\text{II}} \boldsymbol{r} \} + B\_{n}^{\text{II}} Y\_{0}^{\prime} \{ \boldsymbol{\chi}\_{n}^{\text{II}} \boldsymbol{r} \} \right) \exp\boldsymbol{j}(\boldsymbol{\omega}\boldsymbol{t} - \boldsymbol{\beta}\_{n} \boldsymbol{z}) \tag{4}$$

where J<sup>0</sup> and Y<sup>0</sup> are the zeroth-order Bessel functions of the first and second kinds, respectively. Prime with a function represents the derivative with respect to its argument. A<sup>I</sup> <sup>n</sup>, AII <sup>n</sup> and BII <sup>n</sup> are the field constants, superscript identifying its value, in different analytical regions. γ<sup>I</sup> <sup>n</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> n � �<sup>1</sup>=<sup>2</sup> h i and <sup>γ</sup>II <sup>n</sup> <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup> n � �<sup>1</sup>=<sup>2</sup> h i are the radial propagation constants in regions I and II, respectively. β<sup>n</sup> and k are the phase and the free-space propagation constants, respectively [7].

#### 2.1.2 Circular waveguide with dielectric coaxial insert

Similar to model-1, this model (model-2) also contains a metallic circular waveguide of inner radius rW, inner wall of which is free from any dielectric. The model includes a coaxial dielectric insert of radius rC and relative permittivity ε<sup>r</sup> for the full length of the waveguide (Figure 3) [7].

For the sake of analysis, the structure may be divided into two analytical regions, the central dielectric filled region I: 0≤r , rC, 0≤z , ∞; and the free-space (dielectric free) region II: rC ≤r , rW, 0≤z , ∞. The relevant (axial magnetic and azimuthal electric) field intensity components may be written same as for model-1 (1)–(4), in which the radial propagation constants γ<sup>I</sup> <sup>n</sup> and γII <sup>n</sup> are interpreted as: γI <sup>n</sup> <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup> n � �<sup>1</sup>=<sup>2</sup> and γII <sup>n</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> n � �<sup>1</sup>=<sup>2</sup> , respectively [7].

#### 2.2 Metal-loaded circular waveguide

In Section 2.1, we have studied the dispersion characteristics of dielectricloaded structures. In the present section, we will explore three variants of all metal-loaded structure: (i) the conventional annular metal disc-loaded circular waveguide (Figure 4), (ii) the interwoven-disc-loaded circular waveguide (Figure 5), and (iii) metal vane-loaded circular waveguide (Figure 6) for their dispersion characteristics and the effect of change of the geometry parameters on these characteristics.

2.2.1 Circular waveguide with annular metal discs

Circular waveguide loaded with metal vanes [16–18].

Figure 4.

Figure 5.

Figure 6.

277

Circular waveguide with annular metal discs [9–13].

DOI: http://dx.doi.org/10.5772/intechopen.82124

Interwoven disc-loaded circular waveguides [2, 9, 14, 15].

In this model (model-3) a circular metallic waveguide of inner radius rW is considered in which annular disc of thickness T, inner radius rD and outer radius rW

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide… DOI: http://dx.doi.org/10.5772/intechopen.82124

Figure 4.

from the waveguide extremes). Thus the radial thickness of the dielectric lining can be calculated as rW � rL. For the sake of analysis, the structure may be divided into

nr � � exp <sup>j</sup> <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>n</sup> ð Þ<sup>z</sup> (1)

nr � � exp <sup>j</sup> <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>n</sup> ð Þ<sup>z</sup> (2)

two analytical regions, the central free-space (dielectric free) region I: 0≤r , rL, 0≤z , ∞; and the dielectric filled region II: rL ≤r , rW, 0≤z , ∞ (Figure 2). The relevant (axial magnetic and azimuthal electric) field intensity

> AI <sup>n</sup> J<sup>0</sup> γ<sup>I</sup>

> > 1 γI n AI n J 0 <sup>0</sup> γ<sup>I</sup>

<sup>n</sup> <sup>r</sup> � � <sup>þ</sup> BII

<sup>n</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>

<sup>n</sup> Y<sup>0</sup> γII

<sup>n</sup> Y<sup>0</sup> <sup>0</sup> γII

where J<sup>0</sup> and Y<sup>0</sup> are the zeroth-order Bessel functions of the first and second kinds, respectively. Prime with a function represents the derivative with respect to its

> n � �<sup>1</sup>=<sup>2</sup> h i

radial propagation constants in regions I and II, respectively. β<sup>n</sup> and k are the phase

Similar to model-1, this model (model-2) also contains a metallic circular waveguide of inner radius rW, inner wall of which is free from any dielectric. The model includes a coaxial dielectric insert of radius rC and relative permittivity ε<sup>r</sup> for the

For the sake of analysis, the structure may be divided into two analytical regions,

the central dielectric filled region I: 0≤r , rC, 0≤z , ∞; and the free-space (dielectric free) region II: rC ≤r , rW, 0≤z , ∞. The relevant (axial magnetic and azimuthal electric) field intensity components may be written same as for model-1

> n � �<sup>1</sup>=<sup>2</sup>

In Section 2.1, we have studied the dispersion characteristics of dielectricloaded structures. In the present section, we will explore three variants of all metal-loaded structure: (i) the conventional annular metal disc-loaded circular waveguide (Figure 4), (ii) the interwoven-disc-loaded circular waveguide (Figure 5), and (iii) metal vane-loaded circular waveguide (Figure 6) for their dispersion characteristics and the effect of change of the geometry parameters on

<sup>n</sup> <sup>r</sup> � � � � exp <sup>j</sup> <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>n</sup> ð Þ<sup>z</sup> (3)

<sup>n</sup> <sup>r</sup> � � � � exp <sup>j</sup> <sup>ω</sup><sup>t</sup> � <sup>β</sup><sup>n</sup> ð Þ<sup>z</sup> (4)

<sup>n</sup> are the field constants, superscript identifying its value, in

<sup>n</sup> and γII

, respectively [7].

<sup>n</sup> <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup>

n � �<sup>1</sup>=<sup>2</sup> h i

<sup>n</sup> are interpreted as:

are the

and γII

þ∞ n¼�∞

components may be written as [7]:

Electromagnetic Materials and Devices

HI <sup>z</sup> ¼ ∑ þ∞ n¼�∞

<sup>θ</sup> ¼ jω μ<sup>0</sup> ∑

AII <sup>n</sup> J<sup>0</sup> γII

and the free-space propagation constants, respectively [7].

2.1.2 Circular waveguide with dielectric coaxial insert

(1)–(4), in which the radial propagation constants γ<sup>I</sup>

<sup>n</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>

full length of the waveguide (Figure 3) [7].

and γII

2.2 Metal-loaded circular waveguide

EI

þ∞ n¼�∞

<sup>n</sup> and BII

1 γII n AII n J 0 <sup>0</sup> γII <sup>n</sup> <sup>r</sup> � � <sup>þ</sup> BII

HII <sup>z</sup> ¼ ∑ þ∞ n¼�∞

<sup>θ</sup> ¼ jω μ<sup>0</sup> ∑

<sup>n</sup>, AII

different analytical regions. γ<sup>I</sup>

In region I:

In region II:

EII

argument. A<sup>I</sup>

γI

276

<sup>n</sup> <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup>

these characteristics.

n � �<sup>1</sup>=<sup>2</sup> Circular waveguide with annular metal discs [9–13].

#### Figure 5.

Interwoven disc-loaded circular waveguides [2, 9, 14, 15].

Figure 6. Circular waveguide loaded with metal vanes [16–18].

#### 2.2.1 Circular waveguide with annular metal discs

In this model (model-3) a circular metallic waveguide of inner radius rW is considered in which annular disc of thickness T, inner radius rD and outer radius rW are arranged periodically with periodicity L. The structure is commonly known as the disc-loaded circular waveguide (conventional) (Figure 4) [2, 5, 8–13]. As the structure is periodic, therefore one period of the structure coupled with Floquet's theorem is sufficient for the analysis of the infinitely long structure [1, 2, 5, 8, 9]. For the sake of analysis, the structure may be divided into two analytical regions, the central free-space (disc free) region I: 0≤r , rD, 0≤z , ∞; and the disc occupied region II: rD ≤r , rW, 0≤ z≤L � T (Figure 4). The disc free and disc occupied regions are assumed to support propagating (traveling) and stationary waves, respectively. The relevant (axial magnetic and azimuthal electric) field intensity components in the region I is given by (1) and (2) and in the region II may be written as [2, 5, 9–13]:

In region II:

$$H\_x^{\text{II}} = \sum\_{m=1}^{\infty} A\_m^{\text{II}} Z\_0 \{ \gamma\_m^{\text{II}} r \} \cdot \exp(j\omega t) \cdot \sin(\beta\_m z) \tag{5}$$

In region II:

EII

where Z<sup>0</sup> γIII

<sup>m</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>

In region I:

In region II:

EII

<sup>q</sup>, AII

where AI

279

HII <sup>z</sup> ¼ ∑ þ∞ v¼0

<sup>v</sup> and BII

<sup>γ</sup>II <sup>∑</sup> þ∞ v¼0

<sup>θ</sup> <sup>¼</sup> <sup>j</sup>ωμ<sup>0</sup>

γII

In region III:

HII <sup>z</sup> ¼ ∑ ∞ m¼1

<sup>θ</sup> ¼ jωμ<sup>0</sup> ∑

p r n o <sup>¼</sup> <sup>J</sup><sup>0</sup> <sup>γ</sup>III

m � �<sup>1</sup>=<sup>2</sup> h i and <sup>γ</sup>III

∞ m¼1

DOI: http://dx.doi.org/10.5772/intechopen.82124

EIII

1 γII m AII m J 0 <sup>0</sup> γII mr � � <sup>þ</sup> BII

HIII <sup>z</sup> ¼ ∑ ∞ p¼1 AIII <sup>p</sup> Z<sup>0</sup> γIII p r

<sup>θ</sup> ¼ jωμ<sup>0</sup> ∑

p r n oY<sup>0</sup>

∞ p¼1

1 γIII p AIII <sup>p</sup> Z<sup>0</sup> <sup>0</sup> γIII

<sup>0</sup> γIII <sup>p</sup> rW n o � <sup>J</sup>

<sup>p</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>

is the modal harmonic number in region III [2, 9, 14, 15].

2.2.3 Circular waveguide loaded with metal vanes

and β<sup>m</sup> in region II are defined in the same manner as for model-3. β<sup>p</sup>

free-space region II between two consecutive metal vanes rV ≤r≤rW,

intensity components in the regions I and II may be written as [16–18]:

<sup>γ</sup><sup>I</sup> <sup>∑</sup> þ∞ q¼�∞

<sup>v</sup> Jv <sup>γ</sup>IIr � � <sup>þ</sup> BII

<sup>v</sup> <sup>γ</sup>IIr � � <sup>þ</sup> BII

AI <sup>q</sup> Jq γ<sup>I</sup>

> AI q J 0 <sup>q</sup> γ<sup>I</sup>

<sup>v</sup> Yv <sup>γ</sup>IIr � � � � cos

<sup>v</sup> Y<sup>0</sup> <sup>v</sup> <sup>γ</sup>IIr � � � � cos

function of first and second kinds, respectively, with their primes representing the

<sup>v</sup> are the field constants; J and Y are the ordinary Bessel

HI <sup>z</sup> ¼ ∑ þ∞ q¼�∞

AII

AII v J 0

EI <sup>θ</sup> <sup>¼</sup> <sup>j</sup>ωμ<sup>0</sup>

AII <sup>m</sup> J<sup>0</sup> γII

mr � � <sup>þ</sup> <sup>B</sup>II

mY<sup>0</sup> γII

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

mY<sup>0</sup> <sup>0</sup> γII

> 0 <sup>0</sup> γIII <sup>p</sup> rW n oY<sup>0</sup> <sup>γ</sup>III

the field constants, superscript identifying its value, in different analytical regions.

regions II and III, respectively. The axial phase propagation constants β<sup>n</sup> in region I

[¼ 2pπ=ð Þ L � T � TBH ] is the axial phase propagation constants in region III; here p

This model (model-5) considers a circular waveguide of radius rW and N number of metal vanes of vane-inner-tip radius rV and vane angle ϕ extending axially over the length of the waveguide arranged on the waveguide wall to maintain the azimuthal periodicity (Figure 6) [16–18]. Clearly, the azimuthal periodicity is 2π=N. For the analysis of the structure, it may be divided into two regions; (i) the central cylindrical vane-free free-space region I: 0≤r , rV, 0≤ θ , 2π, and (ii) the

ϕ , θ , 2π=N(Figure 6). The relevant (axial magnetic and azimuthal electric) field

p

mr � � � � expð Þ <sup>j</sup>ω<sup>t</sup> sin ð Þ <sup>β</sup>mz (7)

mr � � � � expð Þ <sup>j</sup>ω<sup>t</sup> sinð Þ <sup>β</sup>mz (8)

<sup>m</sup> <sup>r</sup> � � expð Þ <sup>j</sup>ω<sup>t</sup> sin <sup>β</sup>pz

p r n o; AII

<sup>r</sup> � � exp ð Þ �jq<sup>θ</sup> (11)

<sup>r</sup> � � exp ð Þ �jq<sup>θ</sup> (12)

vπ θð Þ � ϕ 2π=N � ϕ

vπ θð Þ � ϕ 2π=N � ϕ

� � (13)

� � (14)

� �<sup>1</sup>=<sup>2</sup> � � are the radial propagation constants in

� � (9)

<sup>m</sup>, BII

� �, (10)

<sup>m</sup> and AIII

<sup>p</sup> are

n o expð Þ <sup>j</sup>ω<sup>t</sup> sin <sup>β</sup>pz

$$E\_{\theta}^{\text{II}} = j\alpha\mu\_0 \sum\_{m=1}^{\infty} \frac{1}{\mathcal{I}\_m^{\text{II}}} A\_m^{\text{II}} Z\_0' \{\boldsymbol{\chi}\_m^{\text{II}} r\} \cdot \exp(j\alpha t) \sin(\beta\_m z) \tag{6}$$

where Z<sup>0</sup> γII mr � � <sup>¼</sup> <sup>J</sup><sup>0</sup> <sup>γ</sup>II mr � �Y<sup>0</sup> <sup>0</sup> γII mrW � � � <sup>J</sup> 0 <sup>0</sup> γII mrW � �Y<sup>0</sup> γII mr � �; AII <sup>m</sup> is the field constants, superscript identifying its value, in different analytical regions. γII <sup>m</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> m � �<sup>1</sup>=<sup>2</sup> h i is the radial propagation constant in region II. <sup>β</sup>n, defined as β<sup>n</sup> ¼ β<sup>0</sup> þ 2π n=L, is the axial phase propagation constant in region I with β<sup>0</sup> as the axial phase propagation constant for fundamental space harmonic, and n [= 0, �1, �2, �3, …] as space harmonic number. βm, defined as β<sup>m</sup> ¼ mπ=ð Þ L � T , is the axial propagation constants in region II with m (= 1, 2, 3, …) as the modal harmonic numbers in region II [2, 5, 9–13].

#### 2.2.2 Interwoven-disc-loaded circular waveguide

This model (model-4) differs from the conventional disc-loaded circular waveguide due to different additional disc included in between two identical consecutive discs of conventional disc-loaded circular waveguide [2, 9, 14, 15]. Thus, this model is considered with a circular metallic waveguide of inner radius rW in which annular disc of thickness T, inner radius rD and outer radius rW are arranged periodically with periodicity L. In addition, another annular disc of thickness TBH, inner radius rBH and outer radius rW are also arranged periodically with periodicity L such that the disc of thickness TBH is placed in middle of two identical consecutive discs of thickness T. The structure is known as the interwoven-discloaded circular waveguide. Similar to conventional disc-loaded circular waveguide, this structure is also periodic, therefore considering unit cell of the structure with Floquet's theorem suffices for the analysis of the infinitely long structure. The analytical regions of the model may be considered as: (i) region I: 0 ≤r , rD, 0≤z , ∞; (ii) region II: rD ≤r , rBH, 0≤z≤ L � T; and (iii) region III: rBH ≤r , rW, 0≤z≤ð Þ L � T � TBH =2, where L � T and ð Þ L � T � TBH =2 represent the axial-gaps between two consecutive discs of smaller hole and between discs of bigger and smaller holes (Figure 5).

Similar to the conventional disc-loaded circular waveguide (model-3), it is assumed that the disc free (I) and disc occupied (II and III) regions, respectively, support propagating and standing waves. The relevant (axial magnetic and azimuthal electric) field intensity components in the region I is given by (1) and (2) and in the regions II and III may be written as [2, 9, 14, 15]:

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide… DOI: http://dx.doi.org/10.5772/intechopen.82124

In region II:

are arranged periodically with periodicity L. The structure is commonly known as the disc-loaded circular waveguide (conventional) (Figure 4) [2, 5, 8–13]. As the structure is periodic, therefore one period of the structure coupled with Floquet's theorem is sufficient for the analysis of the infinitely long structure [1, 2, 5, 8, 9]. For the sake of analysis, the structure may be divided into two analytical regions, the central free-space (disc free) region I: 0≤r , rD, 0≤z , ∞; and the disc occupied region II: rD ≤r , rW, 0≤ z≤L � T (Figure 4). The disc free and disc occupied regions are assumed to support propagating (traveling) and stationary waves, respectively. The relevant (axial magnetic and azimuthal electric) field intensity components in the region I is given by (1) and (2) and in the region II may be

mr � � expð Þ <sup>j</sup>ω<sup>t</sup> sinð Þ <sup>β</sup>mz (5)

mr � �; AII

is the radial propagation constant in region II. βn, defined as

mr � � expð Þ <sup>j</sup>ω<sup>t</sup> sinð Þ <sup>β</sup>mz (6)

<sup>m</sup> is the field con-

written as [2, 5, 9–13]: In region II:

where Z<sup>0</sup> γII

<sup>m</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>

γII

HII <sup>z</sup> ¼ ∑ ∞ m¼1 AII mZ<sup>0</sup> γII

<sup>θ</sup> ¼ jωμ<sup>0</sup> ∑

mr � �Y<sup>0</sup>

2.2.2 Interwoven-disc-loaded circular waveguide

∞ m¼1

<sup>0</sup> γII mrW � � � <sup>J</sup>

1 γII m AII mZ<sup>0</sup> <sup>0</sup> γII

stants, superscript identifying its value, in different analytical regions.

0 <sup>0</sup> γII mrW � �Y<sup>0</sup> γII

β<sup>n</sup> ¼ β<sup>0</sup> þ 2π n=L, is the axial phase propagation constant in region I with β<sup>0</sup> as the axial phase propagation constant for fundamental space harmonic, and n [= 0, �1, �2, �3, …] as space harmonic number. βm, defined as β<sup>m</sup> ¼ mπ=ð Þ L � T , is the axial propagation constants in region II with m (= 1, 2, 3, …) as the modal harmonic

This model (model-4) differs from the conventional disc-loaded circular waveguide due to different additional disc included in between two identical consecutive discs of conventional disc-loaded circular waveguide [2, 9, 14, 15]. Thus, this model is considered with a circular metallic waveguide of inner radius rW in which annular disc of thickness T, inner radius rD and outer radius rW are arranged periodically with periodicity L. In addition, another annular disc of thickness TBH, inner radius rBH and outer radius rW are also arranged periodically with periodicity

L such that the disc of thickness TBH is placed in middle of two identical

consecutive discs of thickness T. The structure is known as the interwoven-discloaded circular waveguide. Similar to conventional disc-loaded circular waveguide, this structure is also periodic, therefore considering unit cell of the structure with Floquet's theorem suffices for the analysis of the infinitely long structure. The analytical regions of the model may be considered as: (i) region I: 0 ≤r , rD,

0≤z , ∞; (ii) region II: rD ≤r , rBH, 0≤z≤ L � T; and (iii) region III: rBH ≤r , rW, 0≤z≤ð Þ L � T � TBH =2, where L � T and ð Þ L � T � TBH =2 represent the axial-gaps between two consecutive discs of smaller hole and between discs of bigger and

Similar to the conventional disc-loaded circular waveguide (model-3), it is assumed that the disc free (I) and disc occupied (II and III) regions, respectively, support propagating and standing waves. The relevant (axial magnetic and azimuthal electric) field intensity components in the region I is given by (1) and (2)

and in the regions II and III may be written as [2, 9, 14, 15]:

EII

mr � � <sup>¼</sup> <sup>J</sup><sup>0</sup> <sup>γ</sup>II

Electromagnetic Materials and Devices

numbers in region II [2, 5, 9–13].

smaller holes (Figure 5).

278

m � �<sup>1</sup>=<sup>2</sup> h i

$$H\_x^{\text{II}} = \sum\_{m=1}^{\infty} \left[ A\_m^{\text{II}} J\_0 \{ \chi\_m^{\text{II}} r \} + B\_m^{\text{II}} Y\_0 \{ \chi\_m^{\text{II}} r \} \right] \exp(j\alpha t) \sin \left( \beta\_m \mathbf{z} \right) \tag{7}$$

$$E\_{\theta}^{\text{II}} = j\omega\mu\_{0} \sum\_{m=1}^{\infty} \frac{1}{\mathcal{I}\_{m}^{\text{II}}} \left[ A\_{m}^{\text{II}} J\_{0}^{\prime} \{ \boldsymbol{\gamma}\_{m}^{\text{II}} \boldsymbol{r} \} + B\_{m}^{\text{II}} Y\_{0}^{\prime} \{ \boldsymbol{\gamma}\_{m}^{\text{II}} \boldsymbol{r} \} \right] \, \exp(j\omega\boldsymbol{t}) \, \sin(\beta\_{m}\boldsymbol{z}) \tag{8}$$

In region III:

$$H\_x^{\text{III}} = \sum\_{p=1}^{\infty} A\_p^{\text{III}} Z\_0 \left\{ \gamma\_p^{\text{III}} r \right\} \exp(j\omega t) \sin \left( \beta\_p z \right) \tag{9}$$

$$E\_{\theta}^{\text{III}} = j\omega\mu\_0 \sum\_{p=1}^{\infty} \frac{1}{\mathcal{I}\_p^{\text{III}}} A\_p^{\text{III}} Z\_0' \{\boldsymbol{\chi}\_m^{\text{III}} r\} \, \exp(j\omega t) \, \sin\left(\beta\_p z\right), \tag{10}$$

where Z<sup>0</sup> γIII p r n o <sup>¼</sup> <sup>J</sup><sup>0</sup> <sup>γ</sup>III p r n oY<sup>0</sup> <sup>0</sup> γIII <sup>p</sup> rW n o � <sup>J</sup> 0 <sup>0</sup> γIII <sup>p</sup> rW n oY<sup>0</sup> <sup>γ</sup>III p r n o; AII <sup>m</sup>, BII <sup>m</sup> and AIII <sup>p</sup> are the field constants, superscript identifying its value, in different analytical regions. γII <sup>m</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> m � �<sup>1</sup>=<sup>2</sup> h i and <sup>γ</sup>III <sup>p</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> p � �<sup>1</sup>=<sup>2</sup> � � are the radial propagation constants in regions II and III, respectively. The axial phase propagation constants β<sup>n</sup> in region I and β<sup>m</sup> in region II are defined in the same manner as for model-3. β<sup>p</sup> [¼ 2pπ=ð Þ L � T � TBH ] is the axial phase propagation constants in region III; here p is the modal harmonic number in region III [2, 9, 14, 15].

#### 2.2.3 Circular waveguide loaded with metal vanes

This model (model-5) considers a circular waveguide of radius rW and N number of metal vanes of vane-inner-tip radius rV and vane angle ϕ extending axially over the length of the waveguide arranged on the waveguide wall to maintain the azimuthal periodicity (Figure 6) [16–18]. Clearly, the azimuthal periodicity is 2π=N. For the analysis of the structure, it may be divided into two regions; (i) the central cylindrical vane-free free-space region I: 0≤r , rV, 0≤ θ , 2π, and (ii) the free-space region II between two consecutive metal vanes rV ≤r≤rW, ϕ , θ , 2π=N(Figure 6). The relevant (axial magnetic and azimuthal electric) field intensity components in the regions I and II may be written as [16–18]:

In region I:

$$H\_x^I = \sum\_{q = -\infty}^{+\infty} A\_q^I J\_q\{\chi^I r\} \exp\left(-jq\theta\right) \tag{11}$$

$$E\_{\theta}^{I} = \frac{j a \mu\_0}{\mathcal{Y}^I} \sum\_{q = -\infty}^{+\infty} A\_q^I J\_q' \{ \boldsymbol{\chi}^I \boldsymbol{r} \} \exp\left( -j q \theta \right) \tag{12}$$

In region II:

$$H\_x^{\text{II}} = \sum\_{v=0}^{+\infty} \left[ A\_v^{\text{II}} J\_v \{ \chi^{\text{II}} r \} + B\_v^{\text{II}} Y\_v \{ \chi^{\text{II}} r \} \right] \cos \left( \frac{v \pi \ (\theta - \phi)}{2 \pi / N - \phi} \right) \tag{13}$$

$$E\_{\theta}^{\text{II}} = \frac{j\alpha\mu\_0}{\mathcal{Y}^{\text{II}}} \sum\_{v=0}^{+\infty} \left[ A\_v^{\text{II}} J\_v' \{ \boldsymbol{\chi}^{\text{II}} \boldsymbol{r} \} + B\_v^{\text{II}} Y\_v' \{ \boldsymbol{\chi}^{\text{II}} \boldsymbol{r} \} \right] \cos\left( \frac{v\pi \ (\theta - \phi)}{2\pi/\mathcal{N} - \phi} \right) \tag{14}$$

where AI <sup>q</sup>, AII <sup>v</sup> and BII <sup>v</sup> are the field constants; J and Y are the ordinary Bessel function of first and second kinds, respectively, with their primes representing the derivatives with respect to their arguments. q is an integer; and v is a non-negative integer. <sup>γ</sup><sup>I</sup> <sup>¼</sup> <sup>γ</sup>II <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup> <sup>1</sup>=<sup>2</sup> and <sup>β</sup> are the radial and the axial phase propagation constants, respectively. In order to include the effect of azimuthal harmonics due to angular periodicity of the structure, the azimuthal dependence is considered as exp ð Þ �jvθ , such that v ¼ s þ qN, where s is also an integer [16–18].

The relevant (axial magnetic and azimuthal electric) field intensity components in the region I may be given by (1) and (2), and in regions II and III are written

Metal- and Dielectric-Loaded Waveguide: An Artificial Material for Tailoring the Waveguide…

mY<sup>0</sup> γII

mY<sup>0</sup> <sup>0</sup> γII

> 0 <sup>0</sup> γIII <sup>m</sup> rW � �Y<sup>0</sup> γIII

m � �<sup>1</sup>=<sup>2</sup> h i, and <sup>γ</sup>III

the field constants in different analytical regions, identified by given superscript,

<sup>m</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>

This model (model-8) is similar to model-5 except the region II filled with dielectric of relative permittivity ε<sup>r</sup> between the two consecutive metal vanes [21]. For the sake of analysis, the structure may be divided into two regions; (i) the central cylindrical vane-free free-space region I: 0≤r , rV, 0 ≤θ , 2π, and (ii) the

Circular waveguide loaded with dielectric and metal discs with the hole radius of metal discs lesser than that of

dielectric filled region II between two consecutive metal vanes: rV ≤r≤rW, , θ , 2π=N (Figure 9). The relevant (axial magnetic and azimuthal electric) field intensity components in the regions I and II may be given by (11)–(14), in which

the radial propagation constant is given as: <sup>γ</sup>II <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

the radial propagation constants in regions I, II, and III, respectively (Figure 8). The axial phase propagation constants β<sup>n</sup> in region I and β<sup>m</sup> in regions II and III are

mr � � � � expð Þ <sup>j</sup>ω<sup>t</sup> sinð Þ <sup>β</sup>mz (15)

mr � � � � expð Þ <sup>j</sup>ω<sup>t</sup> sinð Þ <sup>β</sup>mz (16)

<sup>m</sup> <sup>r</sup> � � expð Þ <sup>j</sup>ω<sup>t</sup> sinð Þ <sup>β</sup>mz (17)

<sup>m</sup> <sup>r</sup> � �; AII

<sup>m</sup> <sup>r</sup> � � expð Þ <sup>j</sup>ω<sup>t</sup> sinð Þ <sup>β</sup>mz (18)

<sup>m</sup>, BII

<sup>m</sup> <sup>¼</sup> <sup>ε</sup>rk<sup>2</sup> � <sup>β</sup><sup>2</sup>

[21].

<sup>m</sup> and AIII

m � �<sup>1</sup>=<sup>2</sup> h i are

<sup>m</sup> are

as [20]:

In region II:

EII

where Z<sup>0</sup> γIII

respectively. γ<sup>I</sup>

Figure 8.

281

dielectric discs [20].

In region III:

HII <sup>z</sup> ¼ ∑ ∞ m¼1

> ∞ m¼1

DOI: http://dx.doi.org/10.5772/intechopen.82124

EIII

<sup>m</sup> <sup>r</sup> � � <sup>¼</sup> <sup>J</sup><sup>0</sup> <sup>γ</sup>III

<sup>n</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> � <sup>β</sup><sup>2</sup>

1 γII m AII m J 0 <sup>0</sup> γII mr � � <sup>þ</sup> BII

HIII <sup>z</sup> ¼ ∑ ∞ m¼1 AIII <sup>m</sup> Z<sup>0</sup> γIII

<sup>θ</sup> ¼ jωμ<sup>0</sup> ∑

<sup>m</sup> <sup>r</sup> � �Y<sup>0</sup>

n � �<sup>1</sup>=<sup>2</sup> h i, <sup>γ</sup>II

defined in the same manner as for model-3 [20].

∞ m¼1

> <sup>0</sup> γIII <sup>m</sup> rW � � � <sup>J</sup>

1 γIII m AIII <sup>m</sup> Z<sup>0</sup> <sup>0</sup> γIII

2.3.3 Circular waveguide loaded with alternate dielectric and metal vanes

<sup>θ</sup> ¼ jωμ<sup>0</sup> ∑

AII <sup>m</sup> J<sup>0</sup> γII

mr � � <sup>þ</sup> BII
