Cavity perturbation theory
In mathematics and electronics, cavity perturbation theory describes methods for derivation of perturbation formulae for performance changes of a cavity resonator.
There are many industrial applications for cavity resonators, including microwave ovens, microwave communication systems, and remote imaging systems using electro magnetic waves.
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 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).
It is convenient to denote cavity frequencies with a complex number
Cavity perturbation theory has been initially proposed by Bethe-Schwinger in optics [1] , and Waldron in the radio frequency domain.
[2] These initial approaches rely on formulae that consider stored energy where
are the complex frequencies of the perturbed and unperturbed cavity modes, and
are the electromagnetic fields of the unperturbed mode (permeability change is not considered for simplicity).
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.
For instance, it is apparent that Expression (1) predicts a change of the Q factor (
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 [3] that are complex quantities, the imaginary part being related to the leakage.
In this framework, the frequency shift and the Q change are predicted by The accuracy of the seminal equation 2 has been verified in a variety of complicated geometries.
For low-Q cavities, such as plasmonic nanoresonators that are used for sensing, equation 2 has been shown to predict both the shift and the broadening of the resonance with a high accuracy, whereas equation 1 is inaccurately predicting both.
[4] For high-Q photonic cavities, such as photonic crystal cavities or microrings, experiments have evidenced that equation 2 accurately predicts both the shift and the Q change, whereas equation 1 accurately predicts the shift only.
represent time-average electric and magnetic energies contained in
This expression can also be written in terms of energy densities [7] as: Considerable accuracy improvements of the predictive force of Equation (5) can be gained by incorporating local field corrections,[4] 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.
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 resonators.
[8][9][10][11][12][13] For rectangular waveguide cavity, field distribution of dominant
Ideally, the material to be measured is introduced into the cavity at the position of maximum electric or magnetic field.
In this case, we can use perturbation theory to derive expressions for real and imaginary components of complex material permittivity
represent volumes of original cavity and material sample respectively,
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
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
