In this note the Lagrangian function for the electromagnetic field of a cavity resonator is found. And from this Lagrangian is deduced a perturbation formula which includes Müller's celebrated result as a special case. The same perturbation formula is derived also from the Boltzmann-Ehrenfest adiabatic theorem in a most simple manner

Papas, C. H.

Caltech Authors - Main

ON PERTURBATION THEORY OF ELECTROMAGNETIC CAVITY RESONATORS
by
C. H. Papas
Antenna Laboratory 
Technical Report No. 3
CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California
A Technical Report to the Office of Naval Research 
Prepared for
Office of Naval Research Contract 
Nonr 220(14)
NR 071-262
March 1954
ON PERTURBATION THEORY OF ELECTROMAGNETIC
CAVITY RESONATORS
Abstract
In this note the Lagrangian function for the electromagnetic 
field of a cavity resonator is found. And from this Lagrangian is deduced 
a perturbation formula which includes Muller's celebrated result as a 
special case. The same perturbation formula is derived also from the 
Boltzmann-Ehrenfest adiabatic theorem in a most simple manner.
Introduction
If a cavity resonator has a "simple" shape and is filled with 
a homogeneous, isotropic medium, to calculate its resonant frequencies and 
mode functions is a straightforward task in principle. However, it is some- 
times of interest to determine the oscillatory properties of a cavity that 
differs by a small amount in one or more of its physical characteristics 
from a cavity the oscillatory properties of which are known. For example, 
the calculation of resonant frequency shift accompanying a slight deforma­
tion of the cavity's walls or an introduction of a Small foreign object 
such as a ferrite or semiconductor in the cavity's volume is necessary in 
certain measurement techniques.
We accordingly derive a formula which allows the calculation
of the shift in resonant frequency produced by a foreign body of given size,
shape, dielectric constant, and permeability. We start from the Lagrangian
of the electromagnetic field and deduce a general perturbation relation, a 
special application of which leads to the desired perturbation formula.1
The adiabatic limitations of the perturbation formula are
discussed in the sense of the Boltzmann-Ehrenfest adiabatic invariant
theorem. And it is shown that within the framework of these limitations
the only calculations that have to be made are those of determining the 
magnetization and polarization of a body immersed in a uniform static
field.
FIELDS IN CAVITY RESONATORS OF SIMPLE SHAPE
A cavity resonator is said to have a "simple" shape when each of 
its surfaces is a coordinate surface in a space which allows separation of 
the wave equation. For these simple shapes it is possible to determine the 
resonant frequencies and mode functions explicitly. Since some general 
properties of the cavity fields will be essential to the development of our 
deductions, we shall just briefly review them here.
The electromagnetic field in a lossless cavity (of any shape) 
forms a standing wave. No traveling waves can exist because no energy is 
fed into or extracted from the cavity. Consequently, the electric field 
is everywhere of the same phase. Accordingly, we write the electric field
as
and in consequence of this we can write the magnetic field as
(1)
(2)
where E and H are real vectors. Substituting (1) and (2) into the two 
curl equations of Maxwell, we see that E and H must satisfy the follow- 
ing real, i.e., not complex, equations:
where ℇ and μ are the dielectric constant and permeability of the homo­
geneous, isotropic medium filling the cavity.
-2-
(3)
(4)
(7)
Let us consider the following expression
-3-
It follows from (1) and (2) that the time-average electric and 
magnetic energy densities are respectively given by
(5)
and
(6)
The total electric and magnetic energies are integrals of and
throughout the volume of the cavity and are respectively denoted by
and . As is well-known, at resonance
Lagrangian of the Electromagnetic Field
(8)
where σ, μ, ℇ are respectively the conductivity, permeability, and dielec- 
tric constant of the medium filling the volume V. Je is the electric current 
density and Jm is the magnetic current density. And E and H denote 
the electric and magnetic fields respectively. The harmonic time dependence 
e-iωt has been suppressed.
We recall that the two curl equations of Maxwell are
(9)
(10)
We now compute δL, the first variation of L, produced by the first varia- 
tions, δE and δH, of the electromagnetic field. Taking the first 
variation of (8) we get
We see that the coefficients of δH and δE are zero by virtue of the Max- 
well equations (9) and (10), and hence,
(12)
If n is the outward normal of the surface S bounding the volume V, by 
the divergence theorem (12) becomes
(13)
If S is a perfectly conducting surface, then n x E vanishes on S and 
δL = 0. Or if δH is restricted so that n x δH = 0 on S, again δL = 0. 
If at least one of these conditions is satisfied, then (8) is the Lagrangian 
of the field and (9) and (10) are its Euler-Lagrange equations.
-4-
(11)
With the aid of the vector identity,
we rewrite (11) in the following form:
(11a)
Perturbation Relation
Suppose that volume V is filled with a medium of constitutive 
parameters μa, ℇa, σa. And within this medium we have sources Jaθ and
-5-
Jma oscillating at frequency ωa. Then the corresponding field quantities 
are Ea and Ha. And further suppose that if the volume were filled with 
medium μb, ℇb, σb the sources Jeb and Jmb oscillating at frequency ωb
would give rise to the fields Eb and Hb. In other words, we are consider­
ing the volume V under two different states, "a" and ''b'' . first we consider 
the volume V to be in state a, and we restrict the variations of Ea and
Ha such that
In other words we restrict the variations of the electric and magnetic fields 
of state a to be proportional to the conjugate complex of the fields in 
state b. λ is a proportionality constant. Substituting (14) and (15) into 
(11) we get
(16)
Now consider the volume V to be in state b and restrict the 
variations of Eb and Hb such that δΕb and δHb are proportional to 
the conjugate complex of the fields in state a:
Substituting (17) and (18) in (11) we get
(14)
(15)
(17)
(18)
-6-
We know that the left side of (16) can be written as
(20)
when (15) is substituted into (13); and similarly the left side of (19) can 
be written as
(21)
when (18) is substituted into (13). Consequently, if we add the conjugate 
complex of (19) to (16), we get
This is a most general perturbation relation. We shall use in this note 
only one of its many applications.
We now apply (22) to cavity resonators. Suppose the volume V is 
bounded by a perfectly conducting surface S and within V the dielectric 
constant is ℇa, the permeability is μa, and the conductivity σa is zero. 
The cavity is oscillating at a frequency ωa in a certain mode. Upon intro- 
ducing a lossless foreign body of email volume τ, the resonant frequency of
(19)
(22)
Application of Perturbation Formula to Cavity Resonators
-7-
the cavity changes. We are interested in computing the new resonant fre-
quency.
Since the wall of the cavity is lossless the left side of (22) 
disappears. This can be seen by transforming the volume integral into an 
integral over the surface S and noting that n x E = 0 on perfectly con- 
ducting surface. And since the cavity is assumed to be free of sources, all 
the current densities in (22) are zero. Moreover, the conduct
(22) are zero. Consequently, (22) reduces to
(23)
Ha, Ea, ωa are the fields and the resonant frequency respectively before 
insertion of the foreign body, and μa, ℇa are the constitutive parameters. 
After foreign object of constitutive parameters μb, ℇb is introduced, the 
fields become Eb, Hb and the frequency shifts from ωa to ωb.
We write
(24)
where δωa is the shift in resonant frequency, δℇa is the difference
between the dielectric constant ℇa of the medium and the dielectric cons- 
tant ℇb of the foreign body, δμa is the difference between the permeabi­
lity μa of the medium surrounding the foreign body and the permeability
μb of the foreign body. And similarly δEa and δHa are the perturba­ 
tions in the field due to the foreign body.
Substituting (24) into (23) we get
-8-
(25)
We neglect quantities of the second order, i.e., δωa δμa and δωa δℇa. 
Hence, (25) reduces to
In the first integral the integration extends throughout the volume
of the foreign body only because δμa and δℇa are identically zero out-
side the foreign body. H*a + δH*a is the resultant magnetic field inside
the foreign body and Ha is the undisturbed magnetic field. Similarly,
E*a + δE*a is the electric field within the foreign body and Ea is the
undisturbed electric field. In the second integral we can neglect δH*a 
and δE*a because the volume T of the foreign body, which is the only 
region where the perturbation fields can possibly have an appreciable value, 
is small compared to V. Thus we can rewrite (26) as
(27)
If we consider the case where the medium in the cavity is a 
vacuum and the constitutive parameters of the foreign body are μ1 and
ℇ1, then
(26)
-9-
(28)
where ℇo and μo are the constitutive parameters of vacuum, Ho and Eo 
are the fields in the hollow cavity with no foreign body present. Substitut­
ing (28) into (27), we get
(29)
We can interpret now the integrand of the first integral of (29) in terms of
the polarization P and the magnetization M. We recall that
D = ℇo E + P = ℇE or (ℇ - ℇo) E = P and that B = μoH + M = μ H or
(μ - μo) H = M. In view of this
(30)
(31)
Consequently by substituting (30) into (29) we get
If we define the total magnetization of the foreign body by m and its total 
polarization by p, i.e.,
(32)
(33)
-10-
and if we note that the undisturbed fields, Eo and Ho, are essentially 
homogeneous throughout the small volume τ, (31) becomes
(34)
where
(35)
. Since the
foreign body is small we can assume that its m and p are produced res­
pectively by a magnetostatic field and an electrostatic field.
As an example of the application of (34) let us consider a small 
sphere of dielectric constant ℇ1 and permeability μ1 placed in a cavity
resonator otherwise filled with a vacuum.
Let us first consider the electric ef­
fect, i.e., let us compute p. Since 
the sphere is small (τ << V) we can 
consider the problem as being equal to 
the one of finding p for a dielectric 
sphere in a uniform electrostatic field
Eo. From elementary considerations we know that the resultant electric 
field within the sphere is
is the time-average total energy stored in the cavity before the foreign 
body is introduced.
The relation (34) is very useful and easy to apply. Of course,
the cavity must be of simple shape so that we can determine explicitly the
undisturbed fields Ho and Eo corresponding to a certain mode of fre­
quency ωo as well as the time-average total energy
-11-
and
Since P = D - ℇoE we have
Integrating P throughout τ we obtain
(36)
And proceeding in a similar way for a sphere of permeability μ1 in a mag­
netostatic field Ho we get 
(37)
Substituting (36) and (37) into (34) we get
(38)
where
.The case of a metallic sphere is a limiting case of (38). 
Since a perfectly conducting metal can be considered as having μ = 0 and
ℇ = ∞, we need only put μ1 = 0 and ℇ1 = ∞ in (38) to obtain the fre­
quency shift produced by a perfectly conducting spheres
(39)
-12-
We can qualitatively apply (39 ) to the problem of frequency shifts 
produced by inward and outward dents in the cavity's wall. Suppose at a 
point on the wall, where the magnetic field is zero we produce a small inward 
dent, say of hemispherical shape. Since the magnetic field was originally 
zero in the region occupied by the hemispherical dent, we sot Ho in (39) 
equal to zero. But then the right side of (39 ) is a positive quantity.
This leads us to conclude that the frequency change is negative, i.e., the 
resonant frequency of the cavity decreases. If at this same point (where 
the original magnetic field is zero) we had made an outward dent rather 
than an inward one, the resonant frequency would have increased. This is 
easy to see if we consider the cavity with an outward dent as the original 
cavity and the one with no dent as the perturbed cavity and note that we 
can pass from the original cavity to the perturbed one by an inward dent.
In case a dent is located where the electric field is zero, Eo in (39) 
is equal to zero and the results are just the converse of what they were 
before: where there is no electric field, an inward indentation increases 
the frequency and an outward one decreases it.
The application of (34) to small objects of arbitrary shape amounts
to a computation of the total magnetization and total polarization of the
objects on a static basis. For oblate and prolate spheroids, i.e., for
discs and needles, these computations have been performed and can be found 
in the literature.2
As another illustrative example3 let us consider a cylindrical 
cavity of radius a and length ℓ oscillating in the TM010 mode with a 
thin dielectric rod of radius d along its axis. The question is, what 
is the shift in resonant frequency produced by the presence of the rod?
When no rod is present we know that the field within the cavity is
where k is such that Jo(ka) = 0. The 
first root of the Bessel function is 
2.405 and, hence, k = 2.405/a or
λ = 2π∕k = 2.61a. In the neighborhood
of the axis, i.e., ρ ~ 0, we see from (41) that H0 ̸ is small. Indeed it
vanishes at ρ = 0. On the other hand Ez is maximum at ρ = 0. It is,
therefore, reasonable to assume that the electric effect will be the dominant
one. We accordingly neglect m* ∙ Ho in (34). Thus 
(42)
where we have used the fact that .
But from the first of equations (30) and from equation (33), we know that
(43)
Upon neglecting δEo in (43), we get
(44)
(45)
-13-
(40)
(41)
And it follows from (42) and (44) that
-14-
ℇ1 is the dielectric constant of the rod and may be complex if its conduc­
tivity is different from zero. According to (40) Eo · Eo* = Jo2(kρ). Sub-
stituting this into (45) we find that
(46)
Carrying out the straightforward integrations, we finally obtain an explicit 
formula for the shift in resonant frequency produced by the dielectric rod:
(47)
Adiabatic Considerations
In the foregoing deductions and in the illustrative examples we 
have assumed tacitly that the change of the resonant field is "adiabatic". 
By this we mean that the number of nodal surfaces in the field pattern is 
not changed by the foreign body. If the volume occupied by the foreign 
body is small compared to spatial variations of the field, the undisturbed 
field is homogeneous throughout the volume and the disturbing influence of 
the foreign body is not sufficiently strong to force a jump from the ori­
ginal mode to an adjacent one. Therefore, the restriction that the foreign 
body be small is equivalent to limiting the allowable changes to adiabatic
ones.
It may appear at first that the problem of a dielectric rod in a 
cylindrical cavity violates this restriction. However, this is not the 
case because the TM010 mode does not vary along the rod; it varies only 
across the rod and in this dimension the rod is indeed assumed to be small. 
If, on the other hand, we were to increase the conductivity of the rod, we
-15-
According to the Boltzmann-Ehrenfest4 adiabatic theorem, if the 
state of any oscillating system is changed adiabatically, the product of 
the period and the time average energy remains invariant. That is,
(48)
where C is a constant. It follows from (48) that
(49)
(50)
It is to be noted that (51) is identical to (34).
would find that eventually the rod would be more like a conductor than a 
dielectric, and consequently the original TM010 mode would jump to a coaxial 
line mode.
Application of the Adiabatic Theorem
If we note that is the change in time average energy produced by the
foreign body, then
are the energy changes produced by a
small body placed respectively in a magnetostatic field equal to Ho and 
in an electrostatic field equal to Eo. And finally from (49) and (50) we 
obtain
because
(51)
-16-
1. As was so kindly pointed out by Professor F. Borgnis, this perturba­
tion formula was originally derived in a different manner by Dr. 
Johannes Müller. The reader is referred to Müller's interesting 
paper "Untersuchung uber elektromagnetische Hohlraume'', Hochfrequenz­
technik and Elektroakustik, 54, 157-161 (1939), and to a readable 
paraphrase by H. B. G. Casimir, "Theory of Electromagnetic Waves in 
Resonant Cavities" Philips Research Report, 6, 162-182 (1951).
2. S. A. Schelkunoff and H. T. Friis, "Antennas", p. 580-581, Wiley,
New York, 1952.
W. R. Smythe, "Static and Dynamic Electricity", McGraw-Hill,
New York, 1951.
3. F. Borgnis, Hochfrequenztechnik and Elektroakustic, 59 (1942)
22-26.
4. See page 183-185, in "Les Tenseurs" by L. Brillouin, Dover, New York 
1946.
References :


On Perturbation Theory of Electromagnetic Cavity Resonators

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On Perturbation Theory of Electromagnetic Cavity Resonators

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