A microwave cavity or radio frequency cavity (RF cavity) is a special type of resonator, consisting of a closed (or largely closed) metal structure that confines electromagnetic fields in the microwave or RF region of the spectrum. The structure is either hollow or filled with dielectric material. The microwaves bounce back and forth between the walls of the cavity. At the cavity's resonant frequencies they reinforce to form standing waves in the cavity. Therefore, the cavity functions similarly to an organ pipe or sound box in a musical instrument, oscillating preferentially at a series of frequencies, its resonant frequencies. Thus it can act as a bandpass filter, allowing microwaves of a particular frequency to pass while blocking microwaves at nearby frequencies.
A microwave cavity acts similarly to a resonant circuit with extremely low loss at its frequency of operation, resulting in quality factors (Q factors) up to the order of 106, for copper cavities, compared to 102 for circuits made with separate inductors and capacitors at the same frequency. For superconducting cavities, quality factors up to the order of 1010 are possible. They are used in place of resonant circuits at microwave frequencies, since at these frequencies discrete resonant circuits cannot be built because the values of inductance and capacitance needed are too low. They are used in oscillators and transmitters to create microwave signals, as filters to separate a signal at a given frequency from other signals, wavemeter or frequency meter, Echo Box for pulsed radars to generate artificial targets and to measure the spectrum and transmit frequency, and in microwave relay stations, satellite communications, and microwave ovens.
RF cavities can also manipulate charged particles passing through them by application of acceleration voltage and are thus used in particle accelerators and microwave vacuum tubes such as klystrons and magnetrons.
Theory of operation
Most resonant cavities are made from closed (or short-circuited) sections of waveguide or high-permittivity dielectric material (see dielectric resonator). Electric and magnetic energy is stored in the cavity. This energy decays over time due to several possible loss mechanisms.
The section on 'Physics of SRF cavities' in the article on superconducting radio frequency contains a number of important and useful expressions which apply to any microwave cavity:
The energy stored in the cavity is given by the integral of field energy density over its volume, , where: H is the magnetic field in the cavity and μ0 is the permeability of free space. The power dissipated due just to the resistivity of the cavity's walls is given by the integral of resistive wall losses over its surface, , where: Rs is the surface resistance.
For copper cavities operating near room temperature, Rs is simply determined by the empirically measured bulk electrical conductivity σ see Ramo et al pp.288-289 Simon Ramo, John Roy Whinnery, Theodore Van Duzer (1965). Fields and Waves in Communication Electronics. John Wiley and Sons.
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A resonator's quality factor is defined by , where: ω is the resonant frequency in [rad/s], U is the energy stored in [J], and Pd is the power dissipated in [W] in the cavity to maintain the energy U.
Basic losses are due to finite conductivity of cavity walls and dielectric losses of material filling the cavity. Other loss mechanisms exist in evacuated cavities, for example the multipactor effect or field electron emission. Both multipactor effect and field electron emission generate copious electrons inside the cavity. These electrons are accelerated by the electric field in the cavity and thus extract energy from the stored energy of the cavity. Eventually the electrons strike the walls of the cavity and lose their energy. In superconducting radio frequency cavities there are additional energy loss mechanisms associated with the deterioration of the electric conductivity of the superconducting surface due to heating or contamination.
Every cavity has numerous resonant frequencies that correspond to electromagnetic field modes satisfying necessary boundary conditions on the walls of the cavity. Because of these boundary conditions that must be satisfied at resonance (tangential electric fields must be zero at cavity walls), at resonance, cavity dimensions must satisfy particular values. Depending on the resonance transverse mode, transverse cavity dimensions may be constrained to expressions related to geometric functions, or to zeros of Bessel functions or their derivatives (see below), depending on the symmetry properties of the cavity's shape. Alternately it follows that cavity length must be an integer multiple of half-wavelength at resonance (see page 451 of Ramo et al). In this case, a resonant cavity can be thought of as a resonance in a short circuited half-wavelength transmission line.
The external dimensions of a cavity can be made considerably smaller at its lowest frequency mode by loading the cavity with either capacitive or inductive elements. Loaded cavities usually have lower symmetries and compromise certain performance indicators, such as the best Q factor. As examples, the reentrant cavityIlan Ben-Zvi, Peter H. Ceperley and H. A. Schwettman, "The Design of Re-Entrant Cavities", Particle Accelerators. 1976, Vol. 7, pp. 125-135, https://cds.cern.ch/record/1021070/files/p125.pdf and helical resonator are capacitive and inductive loaded cavities, respectively.
Multi-cell cavity
Single-cell cavities can be combined in a structure to accelerate particles (such as electrons or ions) more efficiently than a string of independent single cell cavities. The figure from the U.S. Department of Energy shows a multi-cell superconducting cavity in a clean room at Fermi National Accelerator Laboratory.
Loaded microwave cavities
A microwave cavity has a fundamental mode, which exhibits the lowest resonant frequency of all possible resonant modes. For example, the fundamental mode of a cylindrical cavity is the TM010 mode. For certain applications, there is motivation to reduce the dimensions of the cavity. This can be done by using a loaded cavity, where a capacitive or an inductive load is integrated in the cavity's structure.
The precise resonant frequency of a loaded cavity must be calculated using finite element methods for Maxwell's equations with boundary conditions.
Loaded cavities (or resonators) can also be configured as multi-cell cavities.
Loaded cavities are particularly suited for accelerating low velocity charged particles. This application for many types of loaded cavities. Some common types are:
The reentrant cavityCarter, Richard G.; Feng, Jinjun; Becker, Ulrich (2007). "Calculation of the Properties of Reentrant Cylindrical Cavity Resonators" (PDF). IEEE Transactions on Microwave Theory and Techniques. 55 (12): 2531–2538 – via IEEE Xplore.
The helical resonator The spiral resonatorE. Jaeschke et al., "The Heidelberg 3MV-CW Heavy Ion Postaccelerator Section Using Independently Phased Spiral Resonators" in IEEE Transactions on Nuclear Science, vol. 24, no. 3, pp. 1136-1140, June 1977, doi: 10.1109/TNS.1977.4328874.
The split-ring resonatorK. W. Shepard, J. E. Mercereau and G. J. Dick, "A New Superconducting Heavy Ion Accelerating Structure Using Chemically Polished Lead Surfaces," in IEEE Transactions on Nuclear Science, vol. 22, no. 3, pp. 1179-1182, June 1975, doi: 10.1109/TNS.1975.4327840. The quarter wave resonator The half wave resonator.Delayen, J. R., and J. E. Mercereau. "Cryogenic test of a superconducting half-wave resonator for the acceleration of heavy ions." Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 257.2 (1987): 71-76. A variant of the half-wave resonator is the spoke resonator.
The Radio-frequency quadrupole Compact Crab cavity. Compact crab cavities are an important upgrade for the LHC.Design and vertical tests of double-quarter wave cavity prototypes for the high-luminosity LHC crab cavity system, Design and vertical tests of double-quarter wave cavity prototypes for the high-luminosity LHC crab cavity system S. Verdú-Andr´es et al, Physical Review Accelerators and Beams, 21, 082002 (2018)
The Q factor of a particular mode in a resonant cavity can be calculated. For a cavity with high degrees of symmetry, using analytical expressions of the electric and magnetic field, surface currents in the conducting walls and electric field in dielectric lossy material.John C. Slater (1969). Microwave Electronics. Dover Publications. New York. Chapter IV p. 69. For cavities with arbitrary shapes, finite element methods for Maxwell's equations with boundary conditions must be used. Measurement of the Q of a cavity are done using a Vector Network analyzer (electrical), or in the case of a very high Q by measuring the exponential decay time of the fields, and using the relationship .
The electromagnetic fields in the cavity are excited via external coupling. An external power source is usually coupled to the cavity by a small aperture, a small wire probe or a loop, see page 563 of Ramo et al. External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis, see Montgomery et al page 232.Montgomery, C. G. & Robert H. Dicke & Edward Mills Purcell, Principles of microwave circuits / edited by C.G. Montgomery, R.H. Dicke, E.M. Purcell, Peter Peregrinus on behalf of the Institution of Electrical Engineers, London, U.K., 1987.
Resonant frequencies The resonant frequencies of a cavity are a function of its geometry.
Rectangular cavity
Resonance frequencies of a rectangular microwave cavity for any or resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. This frequency is given at page 546 of Ramo et al:
where is the wavenumber, with , , being the mode numbers and , , being the corresponding dimensions; c is the speed of light in vacuum; and and are relative permeability and permittivity of the cavity filling respectively.
Cylindrical cavity
The field solutions of a cylindrical cavity of length and radius follow from the solutions of a cylindrical waveguide with additional electric boundary conditions at the position of the enclosing plates. The resonance frequencies are different for TE and TM modes.
TM modes See JacksonJohn David Jackson (physicist), Classical Electrodynamics, Wiley (1967) pp.254-255
TE modes See Jackson
Here, denotes the -th zero of the -th Bessel function, and denotes the -th zero of the derivative of the -th Bessel function. and are relative permeability and permittivity respectively.
Quality factor The quality factor of a cavity can be decomposed into three parts, representing different power loss mechanisms.
, resulting from the power loss in the walls which have finite conductivity. The Q of the lowest frequency mode, or "fundamental mode" are calculated, see pp. 541-551 in Ramo et al for a rectangular cavity (Equation 3a) with dimensions and parameters , and the mode of a cylindrical cavity (Equation 3b) with parameters as defined above.
where is the intrinsic impedance of the dielectric, is the surface resistivity of the cavity walls. Note that . , resulting from the power loss in the lossy dielectric material filling the cavity, where is the loss tangent of the dielectric
, resulting from power loss through unclosed surfaces (holes) of the cavity geometry.
Total Q factor of the cavity can be found as in page 567 of Ramo et al
Comparison to LC circuits