4. Waveguides

When research in this area does not involve necessarily open spaces, and transmission losses should be avoided, what can we do to reduce losses, dispersion, and uncontrolled reflections of microwaves while propagating? Or what can be done to control and measure such dispersions, absorptions, and reflections? The answer is whenever possible, guide the microwaves (Figure 9).

It is well known that very good conductors (metallic) reflect electromagnetic waves with a minimum of losses (these losses are due to Joule effect on the free electrons that are within the skin depth only) (see Table 1). And this skin depth is very small (microns, fractions of microns) for good conductors. So, multiple reflections on hollow metallic pipes are preferred choice to deliver microwaves from here to there (Figures 9 and 10).

Since the beginnings of the microwave technology, previous to world war II, it is well known that metallic hollow pipes with internal, mirror-polished walls can sustain propagation of some particular electromagnetic (EM) modes, TM (transverse magnetic), and TE (transverse electric) and cannot sustain other EM modes [27–29, 37, 38].

A universal condition is that one significant dimension, ξ, of the hollow pipe be exactly a multiple of an integer number of half the wavelength of the microwave to be transmitted through it. Hence ξ = nλ/2 determines the "size" of the cross section of a rectangular or a cylindrical waveguide. λ<sup>g</sup> = c/νnm is the wavelength inside the waveguide. All the theory is consequence of the solutions to Maxwell's equations, under boundary conditions at the walls of the mirror-polished metallic surfaces of the microwaveguide. The particular deductions of the mathematical expressions of the valid E and H fields inside the waveguides are involved and lengthy. We give in Table 2 some rectangular waveguides with their band, frequency operation, cutoff frequencies, and internal dimensions (Figure 10). A typical rectangular Q-band waveguide connected to a Q-cylindrical resonant cavity is shown in Figure 10b.

#### Figure 9.

Rectangular waveguides made of very good conducting metals (copper, silver, gold, and brass) sustaining different patterns of E and H modes propagating along the guides. If the guide is perfectly conducting the electric field intensity, E is zero in the conductor, and E is either normal or zero at the surface. For a TE wave, it is shown that ∇H0<sup>z</sup> is tangent to the wall.


#### Table 2.

Some of the most common rectangular waveguides and their frequency ranges and frequency cutoffs and inner dimensions (mm). The waveguide name WR stands for waveguide rectangular, and the number is the inner dimension width of the waveguide in hundredths of an inch (0.01 inch = 0.254 mm). The different microwave bands are given and can be correlated with the bands shown in Figure 1 (Taken from Wikipedia).

Once we know the electromagnetic patterns that can be formed and sustained in hollow metallic pipes, as the one we show in Figure 11, how can we use them to probe material's properties? The fundamental idea of how to measure

#### Figure 10.

A plane electromagnetic wave propagating in a rectangular hollow waveguide. (a) The lines AB and CD are parallel to wave fronts for the wave propagating to the right and upward. Similarly, BC and DE are parallel to wave fronts traveling to the right and downward. The angle α is the angle of incidence; the broken line FCG represents a ray reflected at C. (b) A laboratory Q-band cylindrical waveguide.

#### Figure 11.

Fundamental idea of how to measure electrodynamic properties (ϵ, μ, σ) of matter by making it interact with microwaves inside a waveguide. Measuring transmission, reflection, or dispersion will give so much information on ϵ, μ, and/or σ of the material.

electrodynamic properties (ϵ, μ, σ) of matter by making it interact with microwaves inside a waveguide is as follows. We just put the material specimen of interest inside the waveguide, at some place where the electric field is predominant if the electrodynamic expected response is diamagnetic, ϵ(ω), or where the magnetic field dominates, if the magnetic response, μ(ω), is to be explored. In this way the specimen will get excited electrically or magnetically, and predominantly the response would be ϵ(dielectric) or μ(magnetic) and by electron conductivity if the specimen is conductive, even if it is poor conductor (as ferrites).

The Interaction of Microwaves with Materials of Different Properties DOI: http://dx.doi.org/10.5772/intechopen.83675

To insert the specimen in the location we want, a hole is made on top of the guide and a material (dielectric, ϵ; magnetic, μ; and/or conductor, σ) is introduced; depending on the position of the hole E or H, the material will interact strongly with the E or the H component, see Figure 11, of the microwaves, and reflection, absorption, dispersion, and transmission will occur, and their measurement can be carried out.

Waveguides are used mainly to measure microwave transmission and energy transmission, T. A greater performance in the interaction of microwaves with (ϵ, μ, σ) materials is achieved when electromagnetic resonant cavities house the microwaves and the material to be studied.

### 5. Electromagnetic cavities

Closed metallic boxes are a particular case of a bounded space but are an important one. When the wavelength of a particular microwave (v.gr. 3 cm, 11 cm, or 8 mm) is trapped inside a box made of very good conductors (copper, silver, gold), the microwave bounces back and forth between the walls, and a pattern of standing waves is formed. The energy absorption at the walls is very small, and by virtue of this property, such a box is, really, a container of electromagnetic energy, concentrated electromagnetic energy in a closed, finite space. The same way we store a beverage in an aluminum can, we can store electromagnetic energy in a similar can (see Figure 12).

In electromagnetic cavities, the microwaves inside form maxima and minima at known distances, and the electromagnetic energy is stored efficiently. The main quantities that describe the electromagnetic behavior of a cavity are the standing wave electromagnetic fields E and H that form inside the cavity, its power and its figure of merit, the Q of the cavity. All these quantities are obtained from the E and H solutions to Maxwell's equations under boundary conditions at the internal metallic walls. Each component of the fields ðE; HÞ obeys a homogeneous wave equation, as the ones given before.

They are solved by separation of variables, the boundary conditions are periodic, and the separation constants become integer numbers: (k1, k2, k3) �> (n, l, m) in Cartesian coordinates, or ( χ <sup>0</sup> mn, m, k), where χ <sup>0</sup> mn are the roots of the normal Bessel functions J 0 mnð χ <sup>0</sup> mn;r ¼ aÞ, in spherical coordinates [28, 37, 38]. The

#### Figure 12.

A metallic can is commonly used to contain and to store a beverage. Similarly, a metallic can is used to sustain and store electromagnetic fields in the form of standing waves.

electrodynamic power of the EM fields inside the cavity is given by jE � Hj ¼ S ¼ V:W=vol ¼ Power=u:area, and the power absorbed by a purely magnetic material is <sup>P</sup> <sup>¼</sup> <sup>ω</sup>H2 <sup>1</sup>χ".

If we are interested in the quantitative study of microwave dispersion, absorption, and reflection by some kind of materials, we produce standing wave patterns inside a resonant cavity, put inside the material to be studied, let the microwaves interact with it, and then measure its absorption by its reflection and its frequency shift (dispersion) by the change of the energy in the cavity (through the Q of the cavity) without and with material sample.

Examples of real microwave laboratory X-band and Q-band waveguides and cavities are as follows: the schematic representations of E and H field patterns in Figures 9, 10 and 11 are propagated in real waveguides shown in Figure 13. This equipment performs high precision microwave measurements; typically the microwaves are combined with static magnetic fields to excite and saturate the magnetic specimen; hence, electromagnets are part of these equipments. The microwave circuitry includes the source, circulators, attenuators, splitters, and so forth. Some waveguides are shown to be connected through flanges to more waveguides that make 90° turns and then connected to more microwave waveguide "plumbing" until it reaches the heart of the microwave source. At the bottom end, the waveguides terminate in a rectangular cavity that hosts a specimen to be studied. Details of the microwave X-band source box are shown; an isolated open end of another waveguide is also shown. Cavities and waveguides can stand several small holes without degrading their performance; hence, several probes can be inserted. The last panel in Figure 13 shows a rectangular cavity which is also fed in its inside with UV-vis light through an optical fiber that enters the microwave cavity space to excite simultaneously the electronic levels and the electron spins of the atoms of the specimen, the collective magnetization, and/or the domain walls of a ferromagnetic specimen. There are also dual (twin) cavities which simultaneously receive microwaves; one is empty and the other is loaded with a specimen; in real time the different absorption measurements are registered. The fact that cavities and waveguides can stand holes in their walls and insertion of different small measurement, excitation, or conducting devices, multiplies greatly the number of experimentations with ϵ, μ, and σ and electronic and vibronic states that can be performed. Basically, the universal measurement in all these cases would be the reflected microwaves and from them the absorptive characteristics of the sample under study.

#### Figure 13.

Common rectangular X-band waveguides and cavities and microwave source box used in laboratory. (a) typically the microwaves are combined with static H fields Electromagnets are part of these equipments, (b) the inside of a microwave source box that delivers in a precise fashion microwaves from a few microwatts up to 180 mW, (c) view of the open end of a waveguide built with a flange to connect with other waveguides, and (d) a waveguide terminated in a rectangular TE011 cavity which is also fed with UV-Vis light through an optical fiber.

The Interaction of Microwaves with Materials of Different Properties DOI: http://dx.doi.org/10.5772/intechopen.83675

#### Figure 14.

Multiple reflections on the inner metallic wall of a wave guide allow the propagation of microwaves long distances, A, a laboratory Q-Band cylindrical wave guide connected on the upper part to a rectangular portion of another wave guide and connected at the bottom, B, to a cylindrical resonant cavity.

The same kind of experimental setups we just briefly described for X-band waveguides and cavities can be very well carried out at other frequencies with the appropriate K, Ku, Q, L, and S microwave equipment. The hoses that carry water and the electric cables that carry electricity can bend and give ≥90° turns, and pipes can be splitted, reduced in size, and so forth. The same with optical fibers and the same with microwave plumbing. Microwave circulators make the E and H fields go round in a circle and leave at the "aperture" of another piece of waveguide. The coupling of a waveguide with a geometry with another waveguide of another geometry is quite possible as we show in Figure 14 for Q-band waveguides. The cylindrical waveguide connects to the right at the bottom with a cylindrical Q-band cavity, and connects to the left with a rectangular waveguide. The hole to insert a sample is at the center of the top wall (Table 3).

### 5.1 Circular cavity resonators

As a way of example, next we give some cylindrical frequency parameters and some standing wave patters (Table 3) allowed to propagate in these cavities for an

#### Table 3.

Concentrate of the basic properties of some cylindrical cavities that sustain some E and H modes (patterns of standing waves). The blue dashed lines are H lines and the red ones are E lines. The second row shows how they propagate along the waveguide. The field components are shown and parameters like the cut-off frequency, the attenuation due to imperfect conductance, the cut-off wave length are also shown.

air-filled circular cylindrical cavity resonator of radius a and length d. The resonant frequencies are

$$(\mathbf{f\_r})\_{\mathrm{TM}\_{\mathrm{map}}} = \frac{1}{2\pi\sqrt{\epsilon\mu}}\sqrt{\left(\frac{\chi\_{\mathrm{mn}}}{\mathbf{a}}\right)^2 + \left(\frac{\mathbf{p}\pi}{\mathbf{d}}\right)^2} \text{ and } (\mathbf{f\_r})\_{\mathrm{TE\_{\mathrm{map}}}} = \frac{1}{2\pi\sqrt{\epsilon\mu}}\sqrt{\left(\frac{\chi'\_{\mathrm{mn}}}{\mathbf{a}}\right)^2 + \left(\frac{\mathbf{p}\pi}{\mathbf{d}}\right)^2}$$

where the boundary condition at the lateral wall, r = a, imposes JmnðχmnÞ ¼ 0 and the χ <sup>0</sup> mn are the roots n of the m Bessel function; hence J 0 mnðχ <sup>0</sup> mnÞ ¼ 0


.

Above we show just a few roots of Jnðχ Þ and of J 0 <sup>n</sup>ðχ <sup>0</sup> mn mnÞ. For TE001 mode sustained in an empty cavity, m = 0, n = 0, p = 1, so χ <sup>0</sup> mn ¼ 3:832, a = 3.65 cm/2 = 1.825 cm, and d = 4.38 cm with ϵ ¼ ϵ<sup>0</sup> y μ ¼ μ0.

#### 5.2 The quality factor of a resonant cavity

It is a fundamental quantity in the theory and evaluation of microwave cavities. The quality factor of the cavity is defined by

$$\mathbf{Q} = \frac{2\pi (\text{Time} - \text{average energy stored at a resonator frequency})}{\text{Energy dissipated in one period}} \tag{35}$$

Q becomes an extremely useful parameter to measure the performance of a cavity and to make quantitative the losses in it when it is empty and/or when it is loaded with a material sample of interest. The higher the Q, the higher the quality of the metallic cavity as reservoir of electromagnetic energy. A cavity with a Q of 17,000, 33,000, or 100,000 will lose energy in a fraction of 1/17,000 or 1/33,000 or 1/100,000 of the initial energy content per cycle. So, these devices are really very good at storing microwave energy. Notice that the inverse of Q is a measure of those losses:

Q�<sup>1</sup> = Energy loss (absorbed, dissipated) per cycle/2π (time-avarage of energy stored at resonant frequency).

Let's call L the inverse of Q; L = Q�<sup>1</sup> . The theory of cavities finds that there are four types, and only four types, of energy losses: (a) by Joule effect on the conducting walls and just within the skin depth, Lσ, (b) by dielectric losses if a dielectric material, <sup>ϵ</sup><sup>=</sup> <sup>ϵ</sup><sup>0</sup> – <sup>i</sup>ϵ″, is introduced in the cavity, <sup>L</sup>ϵ. This means that the dielectric material absorbs microwaves by virtue of its polarized atoms/molecules, (c) by magnetic losses if <sup>a</sup> magnetic material, <sup>μ</sup> <sup>=</sup> <sup>μ</sup><sup>0</sup> – <sup>i</sup>μ″, is introduced in the cavity, L<sup>μ</sup> [27–33]. This means that the magnetic material absorbs microwaves by virtue of its magnetic moments that precess with friction (damping) according to the Landau-Lifshitz equation of motion [LL], or the magnetic domain walls move back and forth trying to follow H(ω) [30], Lμ, (d) any holes or apertures in the cavity, from which some microwave energy can escape, Lh. And the losses are additive, hence: LQ <sup>¼</sup> <sup>L</sup><sup>σ</sup> <sup>þ</sup> <sup>L</sup><sup>ϵ</sup> <sup>þ</sup> <sup>L</sup><sup>μ</sup> <sup>þ</sup> Lh, in terms of <sup>Q</sup>�<sup>1</sup> becomes <sup>1</sup>=<sup>Q</sup> <sup>¼</sup> <sup>1</sup>=Qσ<sup>þ</sup> <sup>1</sup>=<sup>Q</sup> <sup>ω</sup> <sup>W</sup> <sup>ϵ</sup> <sup>þ</sup> <sup>1</sup>=Q<sup>μ</sup> <sup>þ</sup> <sup>1</sup>=Qh. For an empty cavity: <sup>Q</sup> <sup>¼</sup> where <sup>W</sup> <sup>¼</sup> WE <sup>þ</sup> WH is the PL

The Interaction of Microwaves with Materials of Different Properties DOI: http://dx.doi.org/10.5772/intechopen.83675

#### Figure 15.

A variety of ferrites with different sizes and fabrication procedures absorb energy in similar ways. (a) Profile of absorption contains four distinctive regions. Only one is resonant. (b) The usual representation of the absorption is the derivative of the microwave power with respect to magnetic field. (c) Many other ferrites collapse the yellow region, and the FMR region expands to lower and higher fields.

<sup>ϵ</sup> <sup>Ð</sup> <sup>μ</sup> <sup>Ð</sup> <sup>2</sup> total electric and magnetic microwave energy and WE <sup>¼</sup> <sup>E</sup> <sup>d</sup>τ. <sup>2</sup> <sup>4</sup> j j dτ; WH ¼ <sup>4</sup> j j H At resonance: W ¼ 2WE ¼ 2WH:The power loss per unit area, Lσ, is only due to the <sup>1</sup> <sup>1</sup> conductivity of each wall. In this case Pav <sup>¼</sup> j j<sup>2</sup> <sup>R</sup><sup>σ</sup> <sup>¼</sup> j j<sup>2</sup> <sup>H</sup> <sup>R</sup>σ, where <sup>R</sup><sup>σ</sup> is <sup>a</sup> <sup>2</sup> <sup>J</sup><sup>σ</sup> <sup>2</sup> superficial resistance <sup>R</sup><sup>σ</sup> <sup>¼</sup> <sup>b</sup><sup>ρ</sup> <sup>≈</sup><sup>π</sup> b D ρ <sup>δ</sup>, with b, length of cavity; D, diameter of <sup>π</sup>ðD�δÞ<sup>δ</sup> cylindrical cavity; ρ, resistivity (ρ = 1/σ); σ, conductivity of cavity material; δ, skin qffiffiffiffi qffiffiffiffiffiffi depth <sup>δ</sup> <sup>¼</sup> <sup>2</sup><sup>ρ</sup> <sup>¼</sup> <sup>2</sup> ; <sup>ν</sup>, electromagnetic wave frequency; and Js = |Js|, AC current ωμ ωμσ density generated by microwaves inside the cavity wall, generating a power loss PS. Now, what follows has been found experimentally [39, 40]. If an extra conductor as a wire or a conducting film is introduced in the cavity, a new loss term due to Jex (AC current density generated by microwaves within the skin depth of the extra conductor) appears, Pex. Both terms are of the same type; hence, Pσtotal = P<sup>σ</sup> + Pex, and both power loss integrals are of the same type. P<sup>σ</sup> ¼ ∮ Pav ds, and the new Q is: <sup>Q</sup> <sup>¼</sup> <sup>ω</sup> <sup>W</sup> <sup>¼</sup> <sup>2</sup>π<sup>ν</sup> <sup>W</sup> ds. And so, 1/Q <sup>=</sup> 1/Q<sup>σ</sup> <sup>+</sup> 1/Q ex. If we continue adding lossy objects <sup>P</sup>σþPx <sup>∮</sup> Pav inside a cavity, more power loss terms appear and the total loss would be the sum of each loss, Ptotal = ∑Pi, where Pi = Pσ+P<sup>ϵ</sup> +P<sup>μ</sup> + Ph. This analysis on the Q of a loaded cavity and its losses is very powerful to understand what mechanisms are responsible for the total microwave absorption that a magneto conductor exhibits.

## 5.3 Examples of experimental measurements of microwave absorption by different magnetic and/or conducting materials

The first example is microwave ferrites. Their name clearly indicates the main function they have and have been studied with microwaves since their very invention. Microwave ferrites are crucial elements in microwave measurement equipment itself and in a pleyade of different microwave devices [17]. Its ability to absorb greatly microwaves under very specific circumstances and do not absorb them under other set of circumstances makes these materials highly controllable, and that is what engineering requires [11].

As microwave device it is desirable to have wide yellow and green regions in Figure 15 for passive circulator and isolator operation [11, 24]. Ferrites absorb microwave energy in a resonant fashion and under nonresonant conditions, making these responses a very versatile and manageable material. It is very cheap and easy to fabricate [41].

## 6. Resonant absorption of microwaves

What is microwave energy absorption in a resonant fashion? The phenomenon is really ferromagnetic resonance (FMR). What is the role of microwaves in the ferromagnetic resonance phenomenon? A brief description follows; the most common measurements of the ferrite absorption performance or profile are carried out in equipment as the one shown in Figure 13. In addition to the microwave excitation of the ferrite inside the cavity, an extra static magnetic field, H0, is applied to the ferrite through the magnet poles of the electromagnet also shown in Figure 13a. This is why the cavity is seen located at the center of the magnet poles. This field serves to simplify the magnetic structure of domains of the ferrite, and when 250 mT (2500 Oersters) or more are applied, the domain structure has disappeared, and the material becomes magnetically saturated, and the whole sample has the magnetization value Ms, and this Ms as a whole interacts with the microwave magnetic field and absorbs its energy greatly in the form of ћω = gβH0, in which ћω is the energy of a quantum of the microwave field and the right side gβH0 is the magnetic energy splitting of two consecutive magnetic energy levels. This is the well-known Zemann effect, where β = ћe/2me is Bohr magneton and g is the spectroscopic factor (for ferromagnets and ferrimagnets g is close to 2.00 but always larger). To have ћω = gβH0 means that a stimulated transition between two contiguous energy levels, ΔE=gβH0, is taking place and the energy is provided by photons, hν, with v exactly in the microwave region, of the magnetic component of the microwaves. This is called the resonance condition; it is fully quantum and was discovered without knowing what it was in 1946 by R. Griffiths [34] and explained fully 1 year later by Kittel [32]. When this absorption fullfils the Kittel condition, hν = gβH0, it is resonant absorption of energy (no more, no less, just exactly the energy content in a microwave photon hν), meaning that resonant absorption of microwave (photon) energy is performed by the magnetization of a ferromagnetic specimen. When the atomic magnetic moments or uncoupled electron spins are not governed by the strong magnetic exchange interactions, they do not behave collectively, as a unit, and they behave individually. Such is the case of paramagnetic substances. Each atomic magnetic moment, mi, or electron "spin only", S, can and do absorb microwaves individually obeying also the resonance energy equation, hν = gβH0. In these cases the phenomenon is called electron paramagnetic resonance (EPR) or electron spin resonance (ESR). In any of these cases, energy from the microwaves is absorbed resonantly and very efficiently. So, FMR, EPR, and ESR are techniques that measure very accurately the absorption of microwaves (quanta, hν) in the presence of a static magnetic field, H0, of a magnetic sample located in a microwave cavity.

## 7. Nonresonant absorption of microwaves

When the microwave experimental setup is as described but the equality hν = gβH0 is not fullfiled and absorption of microwaves is still registered, then we have a nonresonant absorption of microwaves, and other dissipative process dynamics are taking place. For example, domain walls can be made to oscillate with an external field, and the motion is dissipative, or some conduction "currents" can be operating. So, the same equipment and experimental setup can measure resonant and nonresonant absorption of microwaves. Measurements of this kind, carried out in equipment as in Figure 13a on ferrites fabricated with different methods and with different compositions [41], give very frequent absorption profiles as the ones

### The Interaction of Microwaves with Materials of Different Properties DOI: http://dx.doi.org/10.5772/intechopen.83675

shown in Figure 15. This contains nonresonant and resonant absorption of microwaves. This kind of absorption profile has been known for many years. Here we want to demonstrate that resonant and nonresonant absorption of microwaves coexists in just one measurement that is capturing different microscopic absorption mechanisms at different values of H0. The information obtained this way is very rich. For passive microwave circulators and isolators, it is highly convenient that the regions below resonance (B/R, yellow) and above resonance (A/R, green) be as wide as possible since no absorption is demanded. The widening of these yellow and green regions is a continuous search by modifying ferrite fabrication parameters and continuously measuring this kind of microwave absorption. Yet, many ferrites do the contrary and absorb in all regions of H0. On the other hand, the maximum possible absorption is required in order to sensing it from a distance. In a sense a kind of sink is desirable, like an antenna that works by absorbing greatly microwaves. Much the same way radar works. In order to develop potential applications as the one illustrated in Figure 3, "sensors" that absorb greatly microwaves in preferred directions are required. Some promising materials are Fe79B10Si11 glasscovered amorphous-conducting magnetic microwires (simply FeBSi wires) because they have shown great capacity to absorb microwaves at X-band in an anisotropic fashion [42]. The proposed application in Figure 3 demands a great global absorption of microwaves in order to detect reflected microwaves from implanted magnetic microwires (glass-covered for them to be biologically inert) in patients that have undergone some kind of orthopedic surgery at the level of knee, shoulder, vertebra, hip, and so on. The microwires are implanted with some specific orientation, and as recovery develops and bone grows, or fractures heal, the microwires would move when pushed by the new processes taking place. Those changes are expected to be informative to the surgeons. The idea of the detection is quite similar as how radar detects moving or static "objects" at a distance. The same idea is used in how the laser gun works detecting a speeding vehicle. A good level of reflected, or perturbed. microwave "signal" coming back to the transducer is required. In laboratory models, FeBSi wires have shown great microwave absorption at some particular orientations. Experiments are carried out with the wires inside microwave cavities in equipment as the one shown in Figure 13. Conditions are established for FMR absorption because it is the maximum possible; hence in addition to the X-band microwaves fed to the resonant cavity, an extra static magnetic field is also applied. The physical interactions are as described for ferrites, except that these are amorphous and no long-range order exists and the strong

#### Figure 16.

FMR absorption of microwaves by glass-coated amorphous-conducting ferromagnetic microwires. (a) First derivative of absorption intensity as function of delivered microwave power and (b) the total absorption is a fast-increasing function of power. The Abs% grows to over 1000% for P = 180 mW. The measured absorption for the military material "mu" is shown for comparison, it does not show such effect [42].

crystalline anisotropy does not exist and Hint that goes in Heff is different from the Hint present in ferrites. A typical absorption profile is shown in Figure 16. Here the quantity of interest is the integral of the absorption curve since it gives directly the total microwave power absorbed by the sample. The higher the integral, the better in order to use it as sensor-detector of microwaves. The total absorption resulted in a fast-increasing monotonic function of power. The Abs% grows to over 1000% for P = 180 mW. The measured absorption for the military material "mu" is shown for comparison; it does not show such effect. This is a kind of amplification effect [42].

In, yet, another application, the cavity microwave magnetic field near an extra inserted conducting perturbation is greatly enhanced. Nanomagnets and micromagnets could require for their study an enhancement of the fields they experience inside a resonant cavity. This could be achieved by introducing an extra conductor (wire) in the cavity in order to expose its free electrons to the microwave electric field in the cavity. Induced currents in the conductor, of the same frequency of the microwaves, would produce an extra magnetic field H+ in some small regions, γ, very close to the extra wires, w+ . Placing a micron- or nano-sized sample, η, in region γ of increased field Hincr = H0 + Hint + H+ could produce an amplified ferromagnetic resonance absorption, since now ћω = gβH0 is fulfilled as before, but more microwave effective power is absorbed by a micro- or nano-magnetic material with a not so large total magnetic moment, M, placed at the γ region. This was proven experimentally by R. Rodbell in 1952 [39, 40] more than half a century ago; we consider it a classic of deep understanding of electrodynamics in cavities and a good example of how to use them in novel ways. Present-day microwave experiments of these kinds can be performed on ferromagnetic resonance equipment that looks like the one shown in Figure 13.

One interesting result obtained by Rodbell is that the microwave magnetic field strength near the surface of a conducting rod may be easily made to exceed the maximum magnetic field strength existing in the unperturbed cavity at the same

#### Figure 17.

The intensity of the ferromagnetic resonance measured for a sample of MnFe2O powder as it depends upon position of the sample in a rectangular TE011 X-band microwave cavity. The incident power is 50 mW. (a) FMR with the ferrite alone, (b) ferrite near a cooper wire, (c) ferrite powder glued to the copper wire. The FMR absorption increase amplification is more than 1000X.

### The Interaction of Microwaves with Materials of Different Properties DOI: http://dx.doi.org/10.5772/intechopen.83675

incident power level. This would appear to be a useful means of effectively coupling microwave energy into a magnetic specimen. The experiment described by Rodbell is the measurement of the relative intensity of the microwave magnetic resonance absorption of an MnFe2O2 powder specimen (manganese ferrite) as it depends upon position within a rectangular (TE) microwave cavity at ≈ 9 kMc=s, Figure 17. The powder specimen is cemented onto the outside surface of a quartz capillary tube that is 3 mm long and of 0:25 mmo:d: The capillary is attached to a quartz post so that it can be positioned along the central "E" plane of the cavity with the capillary axis along the microwave E field. The total resonance absorption here is composed of contributions from many, essentially isolated, randomly oriented particles of the ferrite powder; the line width of the composite absorption is about 1000 Oe. The resonance absorption is used here as an indication of the square of the microwave magnetic field strength averaged over the sample. This is the variation expected for a magnetic resonance absorption that is driven by the usual microwave magnetic field strength for this cavity geometry. The experiment is now repeated after introducing into the capillary tube a bare copper wire that is 3 mm long and of 0:025 mmdiam. The cavity coupling and incident microwave power are constants of the experiment. The small variation of the cavity Q with dc magnetic field is used as a measure of the magnetic resonance absorption in the standard way.

The results indicate that the electric field "drives" the absorption; that is, the microwave electric field is locally perturbed and gives rise to a locally large magnetic field. Further confirmation is found in curve (c) of Figure 17 which displays the result of an experiment in which a copper wire of the same size as in (b) is coated directly (no intervening quartz capillary) with approximately the same amount of the ferrite (MnFe2O4) powder. The increased absorption, here relative to (b), is interpreted to be the result of the larger microwave magnetic field that occurs closer to the perturbing conductor.
