In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
,
Cavity
Cavity may refer to:
Biology and healthcare
*Body cavity, a fluid-filled space in many animals where organs typically develop
**Gastrovascular cavity, the primary organ of digestion and circulation in cnidarians and flatworms
*Dental cavity or too ...
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
describes methods for derivation of
perturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbat ...
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
e for performance changes of a cavity resonator.
These performance changes are assumed to be caused by either introduction of a small foreign object into the cavity, or a small deformation of its boundary. Various mathematical methods can be used to study the characteristics of cavities, which are important in the field of microwave systems, and more generally in the field of electro magnetism.
There are many industrial applications for cavity resonators, including microwave ovens, microwave communication systems, and remote imaging systems using electro magnetic waves. How a resonant cavity performs can affect the amount of energy that is required to make it resonate, or the relative stability or instability of the system.
Introduction
When a resonant cavity is perturbed, e.g. by introducing a foreign object with distinct material properties into the cavity or when the shape of the cavity is changed slightly,
electromagnetic fields
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical co ...
inside the cavity change accordingly. This means that all the resonant modes (i.e. the
quasinormal mode
Quasinormal modes (QNM) are the modes of energy dissipation of a perturbed object or field, ''i.e.'' they describe perturbations of a field that decay in time.
Example
A familiar example is the perturbation (gentle tap) of a wine glass with a kni ...
) of the unperturbed cavity slightly change.
Analytically predicting how the perturbation changes the optical response is a classical problem in electromagnetics, with important implications spanning from the radio-frequency domain to present-day nano-optics. The underlying assumption of cavity perturbation theory is that electromagnetic fields inside the cavity after the change differ by a very small amount from the fields before the change. Then
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
for original and perturbed cavities can be used to derive analytical expressions for the resulting resonant frequency shift and linewidth change (or
Q factor change) by referring only to the original unperturbed mode (not the perturbed one).
General theory
It is convenient to denote cavity frequencies with a complex number
, where
is the
angular resonant frequency and
is the inverse of the mode lifetime. Cavity perturbation theory has been initially proposed by Bethe-Schwinger in optics
, and Waldron in the radio frequency domain. These initial approaches rely on formulae that consider stored energy
where
and
are the complex frequencies of the perturbed and unperturbed cavity modes, and
and
are the electromagnetic fields of the unperturbed mode (permeability change is not considered for simplicity). Expression () relies on stored energy considerations. The latter are intuitive since common sense dictates that the maximum change in resonant frequency occurs when the perturbation is placed at the intensity maximum of the cavity mode. However energy consideration in electromagnetism is only valid for Hermitian systems for which energy is conserved. For cavities, energy is conserved only in the limit of very small leakage (infinite Q’s), so that Expression () is only valid in this limit. For instance, it is apparent that Expression () predicts a change of the Q factor (
) only if
is complex, i.e. only if the perturber is absorbent. Clearly this is not the case and it is well known that a dielectric perturbation may either increase or decrease the Q factor.
The problems stems from the fact that a cavity is an open non-Hermitian system with leakage and absorption. The theory of non-Hermitian electromagnetic systems abandons energy, i.e.
products, and rather focuses on
products that are complex quantities, the imaginary part being related to the leakage. To emphasize the difference between the normal modes of Hermitian systems and the resonance modes of leaky systems, the resonance modes are often referred to as
quasinormal mode
Quasinormal modes (QNM) are the modes of energy dissipation of a perturbed object or field, ''i.e.'' they describe perturbations of a field that decay in time.
Example
A familiar example is the perturbation (gentle tap) of a wine glass with a kni ...
. In this framework, the frequency shift and the Q change are predicted by
The accuracy of the seminal equation has been verified in a variety of complicated geometries. For low-Q cavities, such as plasmonic nanoresonators that are used for sensing, equation has been shown to predict both the shift and the broadening of the resonance with a high accuracy, whereas equation is inaccurately predicting both.
For high-Q photonic cavities, such as
photonic crystal
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of Crystal structure, natural crystals gives rise to X-ray crystallograp ...
cavities or microrings, experiments have evidenced that equation accurately predicts both the shift and the Q change, whereas equation only predicts the shift.
The following is written with
products, but would better be understood with
products of
quasinormal mode
Quasinormal modes (QNM) are the modes of energy dissipation of a perturbed object or field, ''i.e.'' they describe perturbations of a field that decay in time.
Example
A familiar example is the perturbation (gentle tap) of a wine glass with a kni ...
theory.
Material perturbation
![Electromagnetic cavity perturbation](https://upload.wikimedia.org/wikipedia/commons/f/f2/Electromagnetic_cavity_perturbation.svg)
When a material within a cavity is changed (
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
and/or
permeability), a corresponding change in resonant frequency can be approximated as:
[ David Pozar, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998.]
where
is the
angular resonant frequency
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
of the perturbed cavity,
is the resonant frequency of the original cavity,
and
represent original
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
respectively,
and
are original
permeability and
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
respectively, while
and
are changes in original permeability and
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
introduced by material change.
Expression () can be rewritten in terms of
stored energies as:
[Mathew, K. T. 2005. Perturbation Theory. Encyclopedia of RF and Microwave Engineering]
where W is the total energy stored in the original cavity and
and
are
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic energy
Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or magnetic moment \mathbf in a magnetic field \mathbf is defined as the mechanical work of the magnetic force (actually magnet ...
densities respectively.
Shape perturbation
![Electromagnetic shape perturbation](https://upload.wikimedia.org/wikipedia/commons/b/b7/Electromagnetic_shape_perturbation.svg)
When a general shape of a resonant cavity is changed, a corresponding change in resonant frequency can be approximated as:
Expression () for change in resonant frequency can additionally be written in terms of time-average stored energies as:
where
and
represent time-average
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particle ...
energies contained in
.
This expression can also be written in terms of energy densities
as:
Considerable accuracy improvements of the predictive force of Equation () can be gained by incorporating local field corrections,
which simply results from the
interface conditions for electromagnetic fields that are different for the displacement-field and electric-field vectors at the shape boundaries.
Applications
Microwave measurement techniques based on cavity perturbation theory are generally used to determine the dielectric and magnetic parameters of materials and various circuit components such as
dielectric resonator
A dielectric resonator is a piece of dielectric ( nonconductive but polarizable) material, usually ceramic, that is designed to function as a resonator for radio waves, generally in the microwave and millimeter wave bands. The microwaves are con ...
s. Since ex-ante knowledge of the resonant frequency, resonant frequency shift and
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
s is necessary in order to extrapolate material properties, these measurement techniques generally make use of standard resonant cavities where resonant frequencies and electromagnetic fields are well known. Two examples of such standard resonant cavities are rectangular and circular
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
cavities and
coaxial cable
Coaxial cable, or coax (pronounced ) is a type of electrical cable consisting of an inner conductor surrounded by a concentric conducting shield, with the two separated by a dielectric ( insulating material); many coaxial cables also have a p ...
s resonators . Cavity perturbation measurement techniques for material characterization are used in many fields ranging from physics and material science to medicine and biology.
[Ogunlade, O.; Yifan Chen; Kosmas, P.; "Measurement of the complex permittivity of microbubbles using a cavity perturbation technique for contrast enhanced ultra-wideband breast cancer detection," Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE, vol., no., pp.6733–6736, Aug. 31 2010-Sept. 4 2010]
Examples
rectangular waveguide cavity
For rectangular waveguide cavity, field distribution of dominant
mode is well known. Ideally, the material to be measured is introduced into the cavity at the position of maximum electric or magnetic field. When the material is introduced at the position of maximum electric field, then the contribution of magnetic field to perturbed frequency shift is very small and can be ignored. In this case, we can use perturbation theory to derive expressions for real and imaginary components of complex material
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
as:
where
and
represent resonant frequencies of original cavity and perturbed cavity respectively,
and
represent volumes of original cavity and material sample respectively,
and
represent
quality factors of original and perturbed cavities respectively.
Once the complex permittivity of the material is known, we can easily calculate its effective
conductivity
Conductivity may refer to:
*Electrical conductivity, a measure of a material's ability to conduct an electric current
**Conductivity (electrolytic), the electrical conductivity of an electrolyte in solution
**Ionic conductivity (solid state), elec ...
and
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
loss tangent
Dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy (e.g. heat). It can be parameterized in terms of either the loss angle ''δ'' or the corresponding loss tangent tan ''δ''. Both refer to the ...
as:
where f is the frequency of interest and
is the free space permittivity.
Similarly, if the material is introduced into the cavity at the position of maximum magnetic field, then the contribution of electric field to perturbed frequency shift is very small and can be ignored. In this case, we can use perturbation theory to derive expressions for complex material
permeability as:
where
is the guide wavelength (calculated as
).
References
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