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Lecture 18

Hollow

Hollow waveguides are useful for high-power . Air has a higher breakdown compared to most materials, and hence, could be a good medium for propagating high power . Also, hollow waveguides are suﬃciently shielded from the rest of the world so that interference from other sources is minimized. Furthermore, for astronomy, they can provide a low-noise system immune to interference. Air generally has less loss than materials, and loss is often the source of thermal noise. A low loss is also a low noise waveguide.1 Many waveguide problems can be solved in closed form. An example is the coaxial waveg- uide previously discussed. But there are many other waveguide problems that have closed form solutions. Closed form solutions to Laplace and Helmholtz equations are obtained by the separation of variables method. The separation of variables method works only for separable coordinate systems. (There are 11 separable coordinates for Helmholtz equations, but 13 for Laplace equation.) Some examples of separable coordinate systems are cartesian, cylindrical, and spherical coordinates. But these three coordinates are about all we need to know for solving many engineering problems. More complicated cases are now handled with numerical methods using computers. When a waveguide has a center conductor or two conductors like a , it can support a TEM wave where both the electric ﬁeld and the magnetic ﬁeld are orthogonal to the direction of propagation. The uniform plane wave is an example of a TEM wave, for instance. However, when the waveguide is hollow or is ﬁlled completely with a homogeneous medium, without a center conductor, it cannot support a TEM mode as we shall prove next. Much of the materials of this lecture can be found in [31, 75, 84].

1There is a ﬂuctuation dissipation theorem [103, 104] that says that when a system loses energy to the environment, it also receives the same amount of energy from the environment in order to conserve energy. Hence, a lossy system loses energy to its environment, but it receives energy back from the environment in terms of thermal noise.

171 172 Electromagnetic Theory 18.1 Hollow Waveguides 18.1.1 Absence of TEM Mode in a Hollow Waveguide

Figure 18.1: Absence of TEM mode in a hollow waveguide enclosed by a PEC wall. The magnetic ﬁeld lines form a closed loop due to the absence of magnetic charges.

We would like to prove by contradiction (reductio ad absurdum) that a hollow waveguide as shown in Figure 18.1 (i.e. without a center conductor) cannot support a TEM mode as follows. If we assume that TEM mode does exist, then the magnetic ﬁeld has to end on itself due to the absence of magnetic charges. It is clear that C Hs · dl 6= 0 about any closed contour following the magnetic ﬁeld lines. But Ampere’s law¸ states that the above is equal to

Hs · dl = jω D · dS + J · dS (18.1.1) ˛C ˆS ˆS

Hence, this equation cannot be satisﬁed unless there are Ez 6= 0 component, or that Jz 6= 0 inside the waveguide. The right-hand side of the above cannot be entirely zero, or this implies that Ez 6= 0 unless a center conductor carrying a current J is there. This implies that a TEM mode in a hollow waveguide without a center conductor cannot exist. Therefore, in a hollow waveguide ﬁlled with homogeneous medium, only TEz or TMz modes can exist like the case of a layered medium. For a TEz wave (or TE wave), Ez = 0, Hz 6= 0 while for a TMz wave (or TM wave), Hz = 0, Ez 6= 0. These classes of problems can be decomposed into two scalar problems like the layerd medium case, by using the pilot potential method. However, when the hollow waveguide is ﬁlled with a center conductor, the TEM mode can exist in addition to TE and TM modes. We will also study some closed form solutions to hollow waveguides, such as the rectan- gular waveguides. These closed form solutions oﬀer us physical insight into the propagation of waves in a hollow waveguide. Another waveguide where closed form solutions can be ob- tained is the circular hollow waveguide. The solutions need to be sought in terms of Bessel functions. Another waveguide with closed form solutions is the elliptical waveguide. However, the solutions are too complicated to be considered. Hollow Waveguides 173

18.1.2 TE Case (Ez = 0, Hz 6= 0)

In this case, the ﬁeld inside the waveguide is TE to z or TEz. To ensure a TE ﬁeld, we can write the E ﬁeld as

E(r) = ∇ × zˆΨh(r) (18.1.2)

Equation (18.1.2) will guarantee that Ez = 0 due to its construction. Here, Ψh(r) is a scalar potential andz ˆ is called the pilot vector.2 The waveguide is assumed source free and ﬁlled with a lossless, homogeneous material. Eq. (18.1.2) also satisﬁes the source-free condition since ∇·E = 0. And hence, from Maxwell’s equations, it can be shown that the electric ﬁeld E(r) satisﬁes the following Helmholtz , or partial diﬀerential equation that

(∇2 + β2)E(r) = 0 (18.1.3) where β2 = ω2µε. Substituting (18.1.2) into (18.1.3), we get

2 2 (∇ + β )∇ × zˆΨh(r) = 0 (18.1.4)

In the above, we assume that ∇2∇ × zˆΨ = ∇ × zˆ(∇2Ψ), or that these operators commute.3 Then it follows that

2 2 ∇ × zˆ(∇ + β )Ψh(r) = 0 (18.1.5)

Thus, if Ψh(r) satisﬁes the following Helmholtz wave equation of partial diﬀerential equa- tion

2 2 (∇ + β )Ψh(r) = 0 (18.1.6) then (18.1.5) is satisﬁed, and so is (18.1.3). Hence, the E ﬁeld constructed with (18.1.2) satisﬁes Maxwell’s equations, if Ψh(r) satisﬁes (18.1.6).

2It “pilots” the ﬁeld so that it is transverse to z. 3This is a mathematical parlance, and a commutator is deﬁned to be [A, B] = AB − BA for two operators A and B. If these two operators commute, then [A, B] = 0. 174 Electromagnetic Field Theory

Figure 18.2: A hollow metallic waveguide with a center conductor (left), and without a center conductor (right).

Next, we look at the boundary condition for Ψh(r) which is derivable from the boundary condition for E. The boundary condition for E is thatn ˆ × E = 0 on C, the PEC wall of the waveguide. But from (18.1.2), using the back-of-the-cab (BOTC) formula,

nˆ × E =n ˆ × (∇ × zˆΨh) = −nˆ · ∇Ψh = 0 (18.1.7)

In applying the BOTC formula, one has to be mindful that ∇ operates on a function to its right, and the function Ψh should be placed to the right of the ∇ operator. ∂ ∂ In the aboven ˆ · ∇ =n ˆ · ∇s where ∇s =x ˆ ∂x +y ˆ∂y sincen ˆ has no z component. The boundary condition (18.1.7) then becomes

nˆ · ∇sΨh = ∂nΨh = 0 on C (18.1.8) which is also known as the homogeneous Neumann boundary condition. Furthermore, in a waveguide, just as in a case, we are looking for traveling solutions of the form exp(∓jβzz) for (18.1.6), or that

z ∓jβz Ψh(r) = Ψhs(rs)e (18.1.9) where rs =xx ˆ +ˆyy, or in short, Ψhs(rs) = Ψhs(x, y). Thus, ∂nΨh = 0 implies that ∂nΨhs = 0. ∂2 2 With this assumption, ∂z2 → −βz , and (18.1.6) becomes even simpler, namely,

2 2 2 2 2 (∇s + β − βz )Ψhs(rs) = (∇s + βs )Ψhs(rs) = 0 , ∂nΨhs(rs) = 0, on C (18.1.10)

2 2 2 where βs = β − βz . The above is a boundary value problem for a 2D waveguide problem. The above 2D wave equation is also called the reduced wave equation. Hollow Waveguides 175

18.1.3 TM Case (Ez 6= 0, Hz = 0) Repeating similar treatment for TM waves, the TM magnetic ﬁeld is then

H = ∇ × zˆΨe(r) (18.1.11) where 2 2 (∇ + β )Ψe(r) = 0 (18.1.12)

We need to derive the boundary condition for Ψe(r) when we know thatn ˆ × E = 0 on the waveguide wall. To this end, we ﬁnd the corresponding E ﬁeld by taking the curl of the magnetic ﬁeld in (18.1.11), and thus the E ﬁeld is proportional to ∂ E ∼ ∇ × ∇ × zˆΨ (r) = ∇∇ · (ˆzΨ ) − ∇2zˆΨ = ∇ Ψ +zβ ˆ 2Ψ (18.1.13) e e e ∂z e e where we have used the BOTC formula to simplify the above. Taking the z component of the above, we get ∂2 E ∼ Ψ + β2Ψ (18.1.14) z ∂z2 e e Assuming that we have a propagating mode inside the waveguide so that

z ∓jβz Ψe ∼ e (18.1.15) 2 2 2 then in (18.1.14), ∂ /∂z → −βz , and 2 2 Ez ∼ (β − βz )Ψe (18.1.16) Therefore, if

Ψe(r) = 0 on C, (18.1.17) then,

Ez(r) = 0 on C (18.1.18) Equation (18.1.16) is also called the homogeneous Dirichlet boundary condition. One can further show from (18.1.13) that the homogeneous Dirichlet boundary condition also implies that the other components of tangential E are zero, namelyn ˆ × E = 0 on the waveguide wall C. Thus, with some manipulation, the boundary value problem related to equation (18.1.12) reduces to a simpler 2D problem, i.e., 2 2 (∇s + βs )Ψes(rs) = 0 (18.1.19) with the homogeneous Dirichlet boundary condition that

Ψes(rs) = 0, rs on C (18.1.20) In the above, we have assumed that

z ∓jβz Ψe(r) = Ψes(rs)e (18.1.21) To illustrate the above theory, we can solve some simple waveguides problems. 176 Electromagnetic Field Theory 18.2 Rectangular Waveguides

Rectangular waveguides are among the simplest waveguides to analyze because closed form solutions exist in cartesian coordinates. One can imagine traveling waves in the xy directions bouncing oﬀ the walls of the waveguide causing standing waves to exist inside the waveguide. As shall be shown, it turns out that not all electromagnetic waves can be guided by a hollow waveguide. Only when the wavelength is short enough, or the is high enough that an electromagnetic wave can be guided by a waveguide. When a waveguide mode cannot propagate in a waveguide, that mode is known to be cut-oﬀ. The concept of cut-oﬀ for hollow waveguide is quite diﬀerent from that of a waveguide we have learned previously.

18.2.1 TE Modes (H Mode or Hz 6= 0 Mode)

For this mode, the scalar potential Ψhs(rs) satisﬁes

∂ (∇ 2 + β 2)Ψ (r ) = 0, Ψ (r ) = 0 on C (18.2.1) s s hs s ∂n hs s

2 2 2 4 where βs = β − βz . A viable solution using separation of variables for Ψhs(x, y) is then

Ψhs(x, y) = A cos(βxx) cos(βyy) (18.2.2)

2 2 2 where βx + βy = βs . One can see that the above is the representation of standing waves in the xy directions. It is quite clear that Ψhs(x, y) satisﬁes equation (18.2.1). Furthermore, cosine functions, rather than sine functions are chosen with the hindsight that the above satisﬁes the homogenous Neumann boundary condition at x = 0, and y = 0 surfaces.

Figure 18.3: The schematic of a rectangular waveguide. By convention, the length of the longer side is usually named a.

4For those who are not familiar with this topic, please consult p. 385 of Kong [31]. Hollow Waveguides 177

To further satisfy the boundary condition at x = a, and y = b surfaces, it is necessary that the boundary condition for eq. (18.1.8) is satisﬁed or that

∂xΨhs(x, y)|x=a ∼ sin(βxa) cos(βyy) = 0, (18.2.3)

∂yΨhs(x, y)|y=b ∼ cos(βxx) sin(βyb) = 0, (18.2.4)

The above puts constraints on βx and βy, implying that βxa = mπ, βyb = nπ where m and n are integers. Hence (18.2.2) becomes mπ  nπ  Ψ (x, y) = A cos x cos y (18.2.5) hs a b where mπ 2 nπ 2 β2 + β2 = + = β2 = β2 − β 2 (18.2.6) x y a b s z Clearly, (18.2.5) satisﬁes the requisite homogeneous Neumann boundary condition at the entire waveguide wall. At this point, it is prudent to stop and ponder on what we have done. Equation (18.2.1) is homomorphic to a matrix eigenvalue problem

A · xi = λixi (18.2.7)

2 where xi is the eigenvector and λi is the eigenvalue. Therefore, βs is actually an eigenvalue, and Ψhs(rs) is an eigenfunction (or an eigenmode), which is analogous to an eigenvector. Here, 2 the eigenvalue βs is indexed by m, n, so is the eigenfunction in (18.2.5). The corresponding eigenmode is also called the TEmn mode. 2 The above condition on βs is also known as the guidance condition for the modes in the waveguide. Furthermore, from (18.2.6), r mπ 2 nπ 2 β = pβ2 − β2 = β2 − − (18.2.8) z s a b And from (18.2.8), when the frequency is low enough, then

mπ 2 nπ 2 β2 = + > β2 = ω2µε (18.2.9) s a b and βz becomes pure imaginary and the mode cannot propagate or become evanescent in the z direction.5 For ﬁxed m and n, the frequency at which the above happens is called the cutoﬀ frequency of the TEmn mode of the waveguide. It is given by r 1 mπ 2 nπ 2 ω = √ + (18.2.10) mn,c µε a b

5We have seen this happening in a medium earlier and also in total internal reﬂection. 178 Electromagnetic Field Theory

When ω < ωmn,c, the TEmn mode is evanescent and cannot propagate inside the waveguide. A corresponding cutoﬀ wavelength is then 2 λmn,c = (18.2.11) m 2 n 2 1/2 [ a + b ]

So when λ > λmn,c, the mode cannot propagate inside the waveguide. When m = n = 0, then Ψh(r) = Ψhs(x, y) exp(∓jβzz) is a function independent of x and y. Then E(r) = ∇ × zˆΨh(r) = ∇s × zˆΨh(r) = 0. It turns out the only way for Hz 6= 0 is for H(r) =zH ˆ 0 which is a static ﬁeld in the waveguide. This is not a very interesting mode, and thus TE00 propagating mode is assumed not to exist and not useful. So the TEmn modes cannot have both m = n = 0. As such, the TE10 mode, when a > b, is the mode with the lowest cutoﬀ frequency or longest cutoﬀ wavelength. For the TE10 mode, for the mode to propagate, from (18.2.11), it is needed that

λ < λ10,c = 2a (18.2.12)

The above has the nice physical meaning that the wavelength has to be smaller than 2a in order for the mode to ﬁt into the waveguide. As a mnemonic, we can think that have “sizes”, corresponding to its wavelength. Only when its wavelength is small enough can the photons go into (or be guided by) the waveguide. The TE10 mode, when a > b, is also the mode with the lowest cutoﬀ frequency or longest cutoﬀ wavelength. It is seen with the above analysis, when the wavelength is short enough, or frequency is high enough, many modes can be guided. Each of these modes has a diﬀerent group and phase velocity. But for most applications, a single guided mode only is desirable. Hence, the knowledge of the cutoﬀ of the fundamental mode (the mode with the lowest cutoﬀ frequency) and the next higher mode is important. This allows one to pick a frequency window within which only a single mode can propagate in the waveguide. It is to be noted that when a mode is cutoﬀ, the ﬁeld is evanescent, and there is no real power ﬂow down the waveguide: Only reactive power is conveyed by such a mode. Bibliography

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