Physics
Resonant Cavity
A resonant cavity is a structure that can store electromagnetic waves by reflecting them back and forth between its walls. This causes the waves to resonate at specific frequencies, creating standing waves within the cavity. Resonant cavities are commonly used in devices such as microwave ovens, lasers, and particle accelerators.
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11 Key excerpts on "Resonant Cavity"
- K. Kurokawa(Author)
- 1969(Publication Date)
- Academic Press(Publisher)
C H A P T E R 4 RESONANT CAVITIES Resonant cavities are devices constructed so that one can utilize resonant phenomena of electromagnetic fields in a space enclosed by good conducting walls. They are useful as circuit elements, particularly as microwave counter-parts of LC resonant circuits in low frequency ranges. Furthermore, their analysis offers an opportunity to demonstrate a powerful eigenfunction approach. For these reasons, this whole chapter is devoted to the theory of resonant cavities. An equivalent circuit of the cavities is obtained and a method to experimentally determine the circuit parameters is studied in detail. In addition, a discussion of cavities with inhomogeneous media is briefly presented. In contrast to waveguides with inhomogeneous media, the completeness of the eigenfunctions can be shown without difficulty. 4.1 Introduction Conductor walls forming a Resonant Cavity generally have finite con-ductives and, hence, introduce losses of electromagnetic energy. Further, in order to utilize the resonances in the cavity, there must be at least one opening through which the inside and outside of the cavity are connected. However, as an idealized model, let us first consider a space completely enclosed by a perfect conductor and study how electromagnetic fields behave in it. Max-well's equations are given by = ; (4.1) = -; (4.2) 170 4A. Introduction 171 Substituting V x (4.2) into (4.1), we have an equation in E alone; V x V x E -ω 2 εμΕ = 0 (in V) (4.3) where an assumption is made that ε and μ are constant in the volume V of the cavity. The boundary condition for E is given by n x E = 0 (on S) (4.4) where n is the unit vector directed outwards normal to the wall surface S. Equations (4.3) and (4.4) constitute another eigenvalue problem whose solutions can exist only when ω 2 εμ = k 2 takes certain discrete values.
eBook - PDFFoundations of Antenna Radiation Theory
Eigenmode Analysis
- Wen Geyi(Author)
- 2023(Publication Date)
- Wiley-IEEE Press(Publisher)
3 Radiation in Cavity Resonator The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. – Sidney Coleman (American physicist, 1937–2007) A resonator is a device that oscillates with greater amplitude at its resonant frequencies. A metal cavity resonator is a hollow metal box and has an infinite number of resonant frequencies, and the corresponding resonant modes form a complete set and may be used to expand the electromagnetic (EM) fields inside the cavity. If the set of the modes is ordered according to increasing resonant fre- quencies, there is always a lowest resonant frequency but no highest one. Hermann Weyl (German mathematician and theoretical physicist, 1885–1955) first investigated the cavity resonator problem by using the theory of linear integral equation and studied the asymptotic distribution of resonant frequencies [1]. The theory of the cavity resonator was then reexamined by a number of authors [e.g. 2–8]. Many books on EM theory and engineering cover the basic theory of cavity resonators [7, 9–23]. In a cavity resonator bounded by a perfect conductor, the vector modal func- tions can be classified into three types and they are generally necessary for the expansion of the fields. The vector modal functions of the first type behave like the TEM modes in waveguide, both their curl and divergence being zero, and they only exist in a multiply connected region, and are called static vector modal functions. The vector modal functions of the second type are similar to the TE modes in a waveguide, with zero divergence and nonzero curl, called divergenceless vector modal functions or natural resonance modes. The vector modal functions of the third type are similar to the TM modes in a wave- guide and have zero curl and nonzero divergence, known as irrotational vector modal functions. 109 Foundations of Antenna Radiation Theory: Eigenmode Analysis, First Edition.
eBook - PDF- Pasquale Maddaloni, Marco Bellini, Paolo De Natale(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
3 Passive resonators A tympanic resonance, so rich and overpowering that it could give an air of verse to a recipe for stewed hare. John McPhee Some people will believe anything if you whisper it to them. Miguel de Unamuno For many contemporary physics experiments, the use of microwave and optical res-onators has become a powerful tool for enhancement in detection sensitivities, nonlinear interactions, and quantum dynamics. Most often, the term cavity is used to describe such electromagnetic resonators. This term has been taken over from microwave technology, where resonators really look like closed cavities, whereas optical resonators traditionally have an open kind of setup. That difference in geometry is related to the fact that optical resonators are usually very large compared with the optical wavelength, whereas microwave cavities are often not much longer than a wavelength. As discussed in the course of this book, microwave and optical cavities allow one to extend the interaction length between matter and field, to build up the optical power, to impose a well-defined mode structure on the electromagnetic field, to implement extreme nonlinear optics, and to study manifestly quantum mechanical behavior associated with the modified vacuum structure and/or the large field associated with a single photon confined to a small volume [100]. Here we shall examine the basic properties of microwave and optical resonators from which we will draw at the appropriate time in the next chapters. We start by considering microwave cavities which represent, in our context, the basis of operation for masers, cesium fountains, and cryogenic sapphire dielectric resonators. Then, we will deal with optical cavities. In a quite general way, these can be defined as arrangements of optical components allowing a beam of light to circulate in a closed path. Such resonators can be made in very different forms.
- Richard G. Carter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
9 3 3 Resonators 3.1 Introduction Resonant cavities are important components in many microwave tubes because of their frequency selective properties and because they store electromagnetic energy. The purpose of this chapter is to describe the features which are common to all types of microwave resonator and to examine the properties of a few important types in more detail. Any closed metal cavity supports an infinite number of electromagnetic reso- nances. In each of these the fields satisfy Maxwell’s equations and the boundary conditions on the metal surface. If resistive losses associated with currents in the walls of the cavity are neglected then the electric and magnetic fields are in phase quadrature. The losses in metallic cavity resonators are usually small so that the fields differ very little from those in a loss-less cavity. The losses can then be cal- culated to a good approximation from the currents flowing in the walls when the losses are neglected. The cavity resonators used in microwave tubes have simple shapes and are commonly cylindrically symmetrical around the axis of the electron beam. The modes which are chiefly of interest are those having a strong axial component of the electric field in the region of the beam. The simplest example is the pill-box cavity whose TM 010 resonance is illustrated in Figure 3.1. This mode can be derived from the TM 01 mode of a circular waveguide by inserting a pair of conducting planes normal to the axis. The resonant frequency for the lowest mode is the cut- off frequency of the waveguide where there is no axial variation of the electric field. Figure 3.1 shows, schematically, the fields, charges and currents in the cavity at intervals of a quarter of the resonant period T 0 . The properties of this cavity are discussed in detail in Section 3.3. For many purposes it is convenient to represent the properties of a cavity resona- tor by its equivalent circuit.
- Ali Akdagli(Author)
- 2011(Publication Date)
- IntechOpen(Publisher)
Radiation losses are eliminated by the use of such closed elements and ohmic loss is reduced because of the large surface areas that are provided for the surface currents. Radio-frequency energy is stored in the resonator fields. The linear dimensions of the usual resonator are of the order of magnitude of the free-space wavelength corresponding to the frequency of excitation. A simple cavity completely enclosed by metallic walls can oscillate in any one of an infinite number of field configurations. The free oscillations are characterized by an infinite number of resonant frequencies corresponding to specific field patterns of modes of oscillation. Among these frequencies there is a smallest one, f c 0 0 λ = (1) , where the free-space wavelength is of the order of magnitude of the linear dimensions of the cavity, and the field pattern is unusually simple; for instance, there are no internal nodes in the electric field and only one surface node in the magnetic field. Behaviour of Electromagnetic Waves in Different Media and Structures 78 The oscillations of such a cavity are damped by energy lost to the walls in the form of heat. This heat comes from the currents circulating in the walls and is due to the finite conductivity of the metal of the walls. The total energy of the oscillations is the integral over the volume of the cavity of the energy density, ( ) 2 2 0 0 1 2 v W E H dv ε μ = + (2) H m 7 0 4 10 / μ π − = × and F m 9 0 1 10 / 36 ε π − = × (3) , where E and H are the electric and magnetic field vectors, in volts/meter and ampere-turns/meter, respectively. The cavity has been assumed to be empty. The total energy W in a particular mode decreases exponentially in time according to the expression, t Q W W e 0 0 ω − = (4) , where f 0 0 2 ω π = and Q is a quality factor of the mode which is defined by ener gy stored in the cavit y Q energy lost in one cycle 2 ( ) ( ) π = . (5) The fields and currents decrease in time with the factor t Q e 0 ω − .
eBook - PDFMicrowave Engineering
Concepts and Fundamentals
- Ahmad Shahid Khan(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Since the basic principles of transmission lines are equally valid to the waveguides, a quar-ter-wave-long waveguide can act as a resonant circuit. Thus, the quarter-wave section of a hollow waveguide can be considered as a Resonant Cavity with similar conditions as applied to the quarter-wave section. In general, any space that is completely enclosed by conducting walls is capable of con-taining oscillating electromagnetic fields. Such an enclosure possessing res-onant properties can be termed as Resonant Cavity or cavity resonator. It is to be mentioned that a low-frequency circuit has only one resonance frequency, whereas a Resonant Cavity can have many resonant frequencies. 140 Microwave Engineering 4.2 Shapes and Types of Cavities The primary frequency of any Resonant Cavity is mainly determined by its physical size and shape. In general, the smaller the cavity, the higher its reso-nant frequency. A cavity can be carved out of a waveguide, a coaxial cable and from a spherical structure. Despite the variation in shapes, the basic principles of operation for all cavities remain the same. 4.2.1 Cavity Shapes Figure 4.2 illustrates some of the commonly used shapes of cavities. Some of these involve more than one geometrical structure. In all these figures, the lengths of cavities are p λ /2, where p = 1, 2, 3, … . Also, their extreme ends are short circuited. 4.2.2 Cavity Types The cavities can be classified in various ways. The first classification is based on shapes of cavities. In the second classification, a cavity may be termed (a) (b) (c) (d) (e) (f) FIGURE 4.2 Different cavity shapes. (a) Rectangular, (b) cubical, (c) cylindrical, (d) spherical, (e) doughnut shaped and (f) cylindrical ring. R C RLC tank circuit Stray inductance L ′ L ′ L ′ L C ′ C ′ C ′ Stray capcitance C ′ L FIGURE 4.1 RLC resonance circuit and its parasitic elements.
eBook - PDF- Andri M. Gretarsson(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
7 Optical Cavities 7.1 Use of Optical Cavities Optical cavities form the core of lasers. As we saw in Chapter 6, a gain medium is placed within an optical cavity in order to get enough optical gain to allow lasing to occur. Optical cavities are also used without a gain medium for their resonant properties; such “external” optical cavities form the core of many of the most sensitive instruments in use today. Due to their high sensitivity, optical cavities are now used very widely. Applications include laser frequency stabilization, laser spectroscopy, gravitational wave detection, quantum mechanics experiments, and nonlinear optics, to name but a few. 7.2 Plane-Wave Cavity The simplest resonant optical cavity consists of two mirrors facing one another. Light en- ters the cavity through one of the mirrors, which is partially transmissive. The distance between the mirrors is fixed so that the light bouncing back and forth between the mirrors executes an integer number of oscillations per round trip and therefore combines construc- tively with light just entering the cavity, thereby forming a driven standing electromagnetic wave. Keeping the optical pathlength between the mirrors at the precise length required for resonance is a significant technical challenge that we will ignore. Also, we don’t address the fact that a real cavity would have finite mirrors. To keep the beam from spilling out of the side of the cavity, one or both of the mirrors would need to be slightly concave. We simply start with the idealized case of infinite flat mirrors and a beam composed of monochromatic infinite plane waves. Figure 7.1 shows the cavity and the amplitudes of the various fields involved. The input beam comes from the laser and impinges on the input mirror. The other mirror is usually known as the end mirror. To minimize optical loss due to the mirror substrates, the input mirror and end mirror are oriented so that the reflective coatings are on the cavity- side of the optics.
eBook - PDF- V P Bykov(Author)
- 1994(Publication Date)
- World Scientific(Publisher)
Chapter 6 ATOMIC RADIATION IN CAVITIES AND WAVEGUIDES In this chapter we continue the investigation of the atomic radiation in a resonant environment. A cavity and waveguide are the resonant atomic environment next in complexity after a free space and unexcited atom. A cavity has a discrete spectrum; numerous experimental and theoretical papers [1-15] are devoted to the investigation of the atomic radiation in cavities. A waveguide has a continuous spectrum. In this respect it is similar to a free space, but for a theoretical investigation it is easier [16], since eigenwaves in a waveguide (in the low frequency part of its spectrum) are going in only one direction. 6.1 Spontaneous emission of an atom in a cavity Many papers are devoted now to the investigation of the atomic radiation in cavities [1-15]. This research field is called now Cavity QED. We limit himself only by basic features of this effect. In the first step we consider general properties of the atomic radiation in a cavity without losses. The cavity without losses has a discrete spectrum. The spectrum of the atom is also discrete if we limit himself by small excitation energies and do not take into account the continuous part of the spectrum, corresponding to high levels of excitation. The interaction of the atom with the cavity electromagnetic field in the first order of the perturbation theory shifts the energy levels of subsystems but the spectrum of the whole system remains discrete. Some stationary states n) of the whole system correspond to the discrete energy levels E n and the arbitrary state of the system can be represented as a superposition of the stationary states m)) = J £C„e-iE t >*n), (1) n in particular, the initial state of the system, in which the atom is excited and the 102 Spontaneous Emission of an Atom in a Cavity 103 photons are absent, can also be represented in such a form.
eBook - PDF- David M. Pozar(Author)
- 2021(Publication Date)
- Wiley(Publisher)
This is also the reason for a loose coupling to the cavity. Resonant Frequencies The geometry of a cylindrical cavity is shown in Figure 6.8. As in the case of the rectangular cavity, the solution is simplified by beginning with the circular waveguide modes, which already satisfy the necessary 6.4 Circular Waveguide Cavity Resonators 259 FIGURE 6.7 | Photograph of a W-band waveguide frequency meter. The knob rotates to change the length of the circular cavity resonator; the scale gives a readout of the frequency. Photograph courtesy of Millitech Inc., Northampton, Mass. boundary conditions on the wall of the circular waveguide. From Table 3.5, the transverse electric fields (E , E ) of the TE nm or TM nm circular waveguide mode can be written as E t (, , z) = e(, ) ( A + e −j nm z + A − e j nm z ) , (6.50) where e(, ) represents the transverse variation of the mode, and A + and A − are arbitrary amplitudes of the forward and backward traveling waves. The propagation constant of the TE nm mode is, from (3.126), nm = √ k 2 − ( p ′ nm a ) 2 , (6.51a) while the propagation constant of the TM nm mode is, from (3.139), nm = √ k 2 − ( p nm a ) 2 , (6.51b) where k = √ . In order to have E t = 0 at z = 0, d, we must choose A + = −A − , and A + sin nm d = 0, or nm d = , for = 0, 1, 2, 3, … , (6.52) which implies that the waveguide must be an integer number of half-guide wavelengths long. Thus, the resonant frequency of the TE nm mode is f nm = c 2 √ r r √ ( p ′ nm a ) 2 + ( d ) 2 , (6.53a) z d E , E z x = 2 a d = 1 FIGURE 6.8 | A cylindrical Resonant Cavity, and the electric field distribution for resonant modes with = 1 or = 2. 260 Chapter 6: Microwave Resonators 0 5 × 10 8 10 × 10 8 15 × 10 8 20 × 10 8 2 4 (2a/d) 2 (2af) 2 , (MHz – cm) 2 6 TM 010 TM 110 TE 111 TM 011 TE 211 TE 112 TM 012 TE 212 TM 112 TE 011 TM 111 FIGURE 6.9 | Resonant mode chart for a cylindrical cavity.
- Georgios Stefanidis, Andrzej Stankiewicz(Authors)
- 2016(Publication Date)
- Royal Society of Chemistry(Publisher)
This is followed by a discussion on novel applicator systems that aim to have better performance in terms of predictability, controlla-bility and efficiency of microwave activation in chemical processes. In this frame, the Labotron system built by Sairem is presented first; this type of equipment uses an internal transmission line to enhance the distribution of the microwave field. Finally, the concept of a coaxial traveling microwave reactor is discussed, which is a novel concept that does away with resonant fields. Several simulation results are presented to illustrate the topics in this chapter; the environment of these simulations is Comsol Multiphysics 3.5 56 and dielectric data were taken from Meredith. 57 4.2 Resonant Microwave Cavities Usually, when microwave heating is applied, it is done by means of an applicator cavity. 58 This is essentially a metal box – a Faraday cage – which contains the microwave field. The walls of this box form one single intercon-nected electrically conducting entity. As opposed to a system with multiple Chapter 4 96 conducting terminals, this means that the cavity can only support electro-magnetic field patterns whose wavelength is short enough to fit inside . Fur-thermore, due to the cavity being fully enclosed, the microwave fields reflect back and forth in it, interfering constructively and destructively in an alter-nating standing wave pattern. The following two facts are essential to under-stand the properties of cavity applicator systems: 1 There is an integer number of wave patterns that can be supported by a cavity. Above a certain wavelength threshold or, equivalently, below a frequency threshold, no field pattern fits; just below the critical wavelength, one wave pattern fits and with decreasing electromagnetic wavelength more wave patterns will fit. 2 The microwave fields resonate. They are a superposition of multiple wavefields traveling in different directions.
eBook - PDF- Jia-Ming Liu(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
6 Optical Resonance 6.1 OPTICAL RESONATOR .............................................................................................................. As discussed in Section 5.3, multiple reflections take place between the two reflective surfaces of a Fabry–Pérot interferometer, resulting in multiple transmitted fields. A transmittance peak occurs when the round-trip phase shift φ RT between the two reflective surfaces is an integral multiple of 2π so that all of the transmitted fields are in phase. From the viewpoint of the field inside the interferometer, this condition results in optical resonance between the two reflective surfaces. Thus a Fabry–Pérot interferometer behaves as an optical resonator, also called a resonant optical cavity. At resonance, the field amplitude inside an optical resonator reaches a peak value due to constructive interference of multiple reflections. The optical energy stored in an optical cavity peaks at its resonance frequencies. An optical cavity can take a variety of forms. Figure 6.1 shows the schematic structures of a few different forms of optical cavities. Though an optical cavity has a clearly defined longitu- dinal axis, the axis can lie on a straight line, as in Fig. 6.1(a), or it can be defined by a folded path, as in Figs. 6.1(b), (c), and (d). A linear cavity defined by two end mirrors, as in Fig. 6.1(a), is known as a Fabry–Pérot cavity because it takes the form of the Fabry–Pérot interferometer. A folded cavity can simply be a folded Fabry–Pérot cavity that supports a standing intracavity field, as in Fig. 6.1(b). A folded cavity can also be a non-Fabry–Pérot ring cavity that supports two independent, contrapropagating intracavity fields, as in Figs. 6.1(c) and (d). An optical cavity provides optical feedback to the optical field in the cavity. Optical resonance occurs when the optical feedback is in phase with the intracavity optical field.
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