UNCHARGED RESONANT SPINNING
The standard oscillation modes of a cavity resonator are quite well known but examination of the equations which define these modes suggests that the equations may also be describing spinning modes as well. As there seems to be no explicit mention of any spinning modes in the literature, further investigation was carried out. In addition to analysing how the equations can apply to both stationary and spinning fields the direction of energy flow, derived from the equations for the spinning cavity field, was examined and confirmed to be rotational. An FDTD (Finite Difference Time Domain) type computer simulation of Maxwell’s equations also showed the fields to be spinning. Finally a practical experiment was conducted using a cylindrical cavity resonant at a frequency of 438MHz and both the conventional stationary and the spinning modes were measured. The importance of spin in particle physics is well recognized and its classical counterpart could be of considerable significance. On this page you will find an initial brief introduction to the conventional cavity modes, knowledge of which is a useful prerequisite to the general discussion of the spinning modes and of the methods used to confirm their existence which follows.
Conventional Fields of a Cavity Resonator
A typical cavity would be a closed rectangular metal box, cylinder or sphere into which an electromagnetic field is in some way introduced. The exact method is unimportant but may be by a short probe wire or loop energized from a radio frequency source of a suitable frequency and protruding into the cavity at a suitable location. Such devices are frequently used as tuned circuits in ultra high frequency (U.H.F.) and microwave devices. It can be shown that the field within the cavity must satisfy a number of conditions which greatly restrict the possible configurations of the field. For example where the electric field meets the metal surface of the cavity it must either reduce to zero or be perpendicular to the metal surface. Any tangential electric field would cause a current to flow in the metal which would neutralize the field. The magnetic field must also go to zero or instead be tangential to the metal surface. These are known as boundary conditions. The other main restriction is that within the interior of the cavity the fields must satisfy Maxwell's equations and there are only a few field configurations which can do this. In fact when all these conditions are met the cavity is resonant and it is found that there are just two main types of resonant modes possible.
For cylindrical cavities one of these has a component of the electric field, but no magnetic field, along the axis of the cylinder. This will be designated the z direction and this is generally assumed to be the direction of wave propagation. This mode is called an E wave or transverse magnetic (abbreviated to TM) wave as the magnetic field is entirely perpendicular (i.e. transverse) to the direction of propagation. The other lowest frequency mode has just a magnetic field in the z direction and is called an H wave or transverse electric (abbreviated TE). The cavity mode designations are based on those existing in waveguides and the cavity is assumed to be formed by blocking off a short section of waveguide. The field directions are therefore the direction of wave propagation in the parent waveguide and although the same designations are used in the cavity the direction of propagation is not necessarily the same in the cavity. For the purpose of naming the modes, however, it will always be assumed that the direction of propagation is the z direction. It is also possible to energize either of these two main cavity modes at higher frequencies and the higher modes can exist in one, two or all three coordinate axis directions. Numbers are usually assigned to designate the modes i.e. 0, 1, 2, 3 etc and they represent harmonics of the mathematical function which describes the wave shape. Three numbers are assigned one for each axis direction and are listed in the order φ, r and z for example TE121.
For a spherical cavity the radial direction is assumed to be the direction of propagation and so a cavity with an H field pointing in the radial direction and with no radial E field is H wave or TE. If no radial H field exists but it has a radial E field it is E wave or TM. Different writers seem to list the mode numbers in different orders but here the order used is θ, r and φ. (Where θ is the angle to the z axis and φ is the angle from the x axis, being positive towards the y axis).
The TM and TE modes for a spherical cavity are illustrated below. Fig 1 shows the electric and magnetic fields with the TM111 mode energized. In these plots the spherical field has been plotted surrounded by an x, y, z coordinate frame box to show the orientation. They do not show conventional field lines as the technique used divides the box into one thousand grid squares and plots an arrow showing the magnitude and direction of the field at each grid line intersection. They do, however, give a reasonable visual indication of the strength and direction of the fields:-
The two fields above are not being viewed from quite the same angle as the orientation used makes it easier to view the field configuration. However, I hope you can see that the magnetic field is of a circular form and the electric field is principally in the direction parallel to the x axis (it does bend outwards at the periphery to ensure it is perpendicular to the walls of the cavity). The electric field is aligned along the axis of the central field minimum of the magnetic field.
The TE mode is:-
One of the important features of all these resonant modes is that they are stationary in space. The fields are varying in amplitude sinusoidally with time but their position does not change. When they reduce to zero they frequently switch polarity so that when they increase they are in the opposite direction. Although this is equivalent to a 180 degree change in orientation it is only a change in polarity of a stationary field. The electric and magnetic field variations are also not in phase. That is to say that when the magnetic field is a maximum the electric field is a minimum so the magnetic field variations are 90 degrees in advance of the electric field.
Maxwell’s equations for the electric and magnetic fields are of a very similar form. It can be seen from the above plots that the TM111 magnetic field is very similar to the TE111 electric field. The principal difference is that the TE111 electric field goes to zero at the cavity wall whereas the TM111 magnetic field need not. This is in order to meet the required boundary conditions. Examining the plots closely it will be found that the central portion of the TM electric field plot is like the TE magnetic field. As the equations for the TE electric field are very similar to those for the TM magnetic field it is the imposition of the different boundary conditions which puts the cavity wall in a different location. Given a cavity of a particular size it follows that the TE mode must be of a higher frequency than the corresponding TM mode to fit the extra field into the cavity.
For a given size of cavity a particular conventional mode will only be supported at one specific resonant frequency. The exact frequency can be found by solving the cavity equations using the field boundary conditions. Although each mode will occur at only one resonant frequency, sometimes two or even three different modes will occur at the same frequency. If multiple resonant modes exist at the same frequency the modes are said to be degenerate. As the energizing frequency is increased there are more and more higher modes that the cavity will support. As the frequency is reduced the lowest frequency mode for a spherical cavity is the TM110 and TM111, which occur together, and if the frequency is less than this no further resonant modes exist. At resonance the fields inside the cavity reach high values. If the cavity is unloaded i.e. there are no outlets for the radio frequency energy, then the fields will increase in magnitude until the losses in the metal of the cavity walls are equal to the power being fed into the cavity. The energy stored in the fields depends on the size of the cavity and the electrical conductivity of the cavity walls and is measured using a term known as Q. This is defined as
Q = ω x energy stored in fields
Power dissipated in fields (this is same as power input)
Where , and f is the cavity resonant frequency. A high Q is an indication of low cavity losses and an 80 cm diameter cavity can have a Q as high as 30,000.
To find the direction of the power flow in the fields of a cavity resonator you just need to work out the direction of the Poynting vector (P). More precisely this gives the rate at which energy crosses a closed surface. The Poynting vector is obtained by taking the vector or cross product of the electric (E) and magnetic (H) fields. The vector cross product of two vectors is written E x H and the result is a vector of magnitude E.H.sinθ, where θ is the angle between E and H, which is therefore a maximum when E and H are at right angles. This vector (P in this case) has a direction perpendicular to the plane of E and H, in the direction an imaginary right handed screw thread would travel if turned in the direction from the first written vector (E) to the second (H). This is illustrated in Fig 3:-
If the Poynting vector for the conventional fields of a cavity resonator is calculated you find that the direction of power flow is radially outward or inward. The direction of energy flow is actually oscillating outward and inward at twice the field frequency. This is similar to the power flows in a standing wave on an aerial feeder or the power flow in a capacitor or inductor connected to an alternating current source. This oscillation occurs because the electric and magnetic fields are in quadrature time phase. If they were in phase the cavity field would always be in just one direction as if both electric and magnetic fields change phase by 180 degrees the Poynting vector does not change direction. However, with phase quadrature just one field will first change phase and this will reverse the direction of the Poynting vector and hence the direction of power flow. The Poynting vector of a conventional spherical cavity resonator mode is plotted in Fig 4 below and this clearly shows the oscillating radial power flow:-
Numerous text books on electromagnetic fields or radio include some cavity resonator theory but Ref 1 lists two books which cover the topic in greater detail.
Spinning Fields of a Cavity Resonator
Everything that has been covered so far is standard text book theory. What I have yet to find mentioned in the literature is the fact that in addition to the above stationary modes, it is also possible to have resonant spinning modes. On this site only cylindrical and spherical cavities are being analysed but spinning modes will be possible in other cavities (such as ellipsoidal) provided they have constant cross section for rotations about the axis of spin. Spinning modes are characterized by the fact that the field is of constant amplitude, not varying with time, but its orientation in space continually changes due to its spin. If a single fixed probe were used to sample the field it would not be possible to distinguish a spinning mode from a fixed mode as the spinning field appears to vary in amplitude as different parts of the field are sampled as it spins. If however two or more probes are used to sample either the electric or magnetic field, then if correctly positioned, it becomes clear that whereas for a stationary mode the sample voltages reach a positive or negative maximum at the same time on each probe for the rotating field the maximum occurs at different times. This depends on when the maximum of the fixed amplitude field passes the probe. Shown below are the field plots for the lowest frequency TM spinning field in a spherical cavity. All spinning plots are shown with the field spinning about the z axis where it passes through the point x=5, y=5 and with an anticlockwise direction of spin when viewed from above. You can confirm this by working out the direction of the Poynting vector:-
There is a TE mode possible which is similar but in which electric and magnetic fields swap positions and the outer boundary is further out than for the TM mode field:-
The rotational frequency of these particular modes is the same as the resonant frequency of the conventional modes. Comparing these plots to those of a conventional cavity it can be seen that the fields are very similar but the alignment between the electric and magnetic fields has rotated 90 degrees about the central z axis. So instead of having a 90 degree time difference in the varying field amplitudes of stationary fields as in the conventional cavity modes we now have constant amplitude but with a 90 degree space difference in spinning fields. Instead of the magnetic field being produced by a change of the electric field with time it is being produced by a change of the electric field in position. Likewise, the electric field is being produced by similar changes in the magnetic field position.
For a cylindrical cavity the lowest modes are TM110 and TE111:-
The above spinning fields are also similar to the conventional fields of a cylindrical cavity but, as before, the electric and magnetic fields are 90 degrees displaced about the central z axis and these fields are of constant amplitude not time varying amplitude.
The field equations for the spinning modes are actually the same as those for the stationary modes but a slightly different interpretation is required of the nature of the phase displacement of the electric and magnetic fields and what the rotational term applies to as indicated above. This is described in more detail and with a little more mathematics on the link Spherical Cavity Field Equations for Stationary and Spinning Modes and the link Cylindrical Cavity Field Equations for Stationary and Spinning Modes. On these links I have restated the equations in a slightly different form to emphasis the rotational aspects of the equations. The equations are non divergent (i.e. the field is not charged) and satisfy all of Maxwell’s equations. The rotational frequency is constant at all distances from the centre axis of the field. This means that the rotational speed increases directly proportional to the distance from the rotation axis of the field. i.e. proportional to ω x r for the cylindrical cavity and as ω x r x sin(θ) for the spherical cavity (where r is the radius, ω is the rotational angular frequency so r x sin(θ ) is the distance from the spherical rotation axis).
The field equations for the spherical cavity show that for the lowest frequency H wave mode (TE111) the tangential electric fields reach their first zero (where the cavity wall must be) when = 4.49. (c is the velocity of light.) The maximum velocity is when θ is 90 degrees in which case ω x r is the tangential velocity. This means that the maximum velocity for the TE111 spherical spinning mode is 4.49 times the velocity of light. This may cause some raised eyebrows but a similar situation arises in waveguide theory. There are two velocities to be considered for an electromagnetic field. The group velocity is the velocity at which a signal could be transmitted and this must be less than c. However, the phase velocity is the velocity at which waves of constant phase travel in an electromagnetic field and this can be far greater than c and this is what the figure of 4.49c refers to. The second radial resonance is a TE121 mode and this occurs when = 7.72 and here, of course, the phase velocity reaches as high as 7.72c.
The two web pages Tables of Spherical Cavity Resonant Modes and Tables of Cylindrical Cavity Resonant Modes list all the lowest stationary and spinning modes possible. In fact each possible stationary mode has a spinning counterpart provided that the fields have a sinusoidal variation in the direction of rotation. This is all modes apart from spherical TE1X0, TE2X0, TM1X0, TM2X0 and cylindrical TE0X1, TM0X0 and TM0X1. (Where X is an integer 1, 2, 3 … etc). Generally the rotation frequency is the same as the stationary resonant frequency. However some fields such as spherical TE21X have X cycles of variation in the φ direction for each rotation and for these the rotation frequency is 1/X times the resonant frequency. The tables include the equations for determining the resonant frequency and calculated resonant frequencies for a specified cavity as an example.
The direction of the Poynting vector for these fields supports the fact that they have a substantial spinning component. If the Poynting vector is plotted for the spinning modes it is found to be tangential to the walls of the cavity. Thus the flow of field energy is in the direction of the Poynting vector which in this case is anticlockwise about the z axis as shown below:-
Confirmation of Spinning Fields
Three different techniques have been used to confirm that the spinning fields can really exist in a cavity. These are:-
1. Theoretical Calculation
The spinning spherical field equations and the spinning cylindrical field equations have been checked to confirm that they comply with Maxwell’s equations for a charge free electric field in vacuum. The calculations are rather tedious and the easiest method to do this is using a symbolic computer mathematics program such as Mathemetica or Maple. Although not included here a program was written using Maple 6 which verified that they comply with the Maxwell’s equations:-
Curl E = - dB/dt and Curl B =
The divergence of the electric field has been computed and is zero, showing that the field is charge free and the divergence of the magnetic field is also zero as required for all magnetic fields. The fact that all components of the field are multiplied by a term of cos(φ-ωt) or sin(φ-ωt) shows that they are all spinning at a rotational frequency of ω in the direction of φ.
2. FDTD Simulation
There is a very powerful computer technique available for determining the behaviour of electromagnetic fields. This is known by the somewhat intimidating name of the Finite Difference Time Domain method or FDTD as it is generally known (Ref 2). This is a simulation which involves setting up a three dimensional grid on which the electric and magnetic field values are entered and then applying Maxwell’s equations to calculate the field values a short time later at each point on the grid. By repeating this many times the computer is able to calculate the exact way the field would evolve. Such a program was run on a spherical TE111 field, a cylindrical TE111 field and a cylindrical TM110 field and all of these were seen to be spinning at the correct theoretical speed. For example the cylindrical TM110 simulation used a three dimensional 24x24x24 grid and a time factor such that a field in free space would cross one grid square in three time steps. The initial cylindrical field had a diameter of 20 grid squares (so a circumference of 62.8 grid lengths) and a height of 20 grid squares. It was a relatively simple program and the simulated cylindrical cavity wall was not smooth but followed the nearest grid square. This gives a rather jagged surface representation but still did not significantly distort the bulk of the field. The program showed that the field completed 31 complete rotations in 1492 time steps i.e. 48.13 time steps per revolution. In one revolution the field would only have covered 48.13/3 = 16.04 grid lengths if travelling at the velocity of light (c) so the perimeter is travelling at 62.8/16.04 = 3.91c. This is within 2.1% of the theoretical figure of 3.83c. Fig 10 is some plots of the magnetic field from this cylindrical TM110, FDTD simulation. The plots are sections of the magnetic field taken at the mid height of the cylinder as this most clearly shows the field rotation which is anticlockwise about the z axis and centered on the x = 12 and y = 12 points. The jagged cylindrical surface used is visible where there is missing field at certain locations around the periphery. As the magnetic field tries to align itself parallel to the cylinder wall this jaggedness is responsible for some of the peripheral fields not being fully aligned with the rest of the spinning field. However, the plots clearly show how the bulk of the field maintains its configuration as it rotates.
The simulation was run for 4400 time steps and fig 11 is a plot of the electric and magnetic fields after 91 revolutions. This again confirms that the fields are largely maintaining their original distribution as they spin.
3. Tests on a Cylindrical Cavity
The fields are rotating at relativistic speeds and have energy and therefore, in the laboratory frame, have mass. It was, therefore, a concern that despite having used two independent theoretical methods for calculating the form of the spinning fields the methods used, both based purely on Maxwell’s equations, may not have taken into account all relevant effects. As a final confirmation a spinning field was therefore produced in an 83.4 cm diameter by 50 cm high cylindrical cavity which was constructed from 18swg copper sheet and energized at about 438MHz in the TM110 mode. Two small single turn coils were used spaced 90 degrees apart about a diameter of the cylinder and at mid height. They were near the wall of the cavity were the Hφ field is a maximum and energized through a phasing line intended to give a 90 degree phase shift between the energizing currents supplying the two coils. The electric field was measured using a four channel oscilloscope which could be connected via isolating resistors to a number of 2.5cm long probe wires projecting through the top cylinder end plate.
The circularity of the cavity was found to be very critical to obtain the correct phasing of the energizing coils but when correctly adjusted the amplitude of the voltage at each of the probes was very similar and the phase difference of the voltage at probes spaced φ degrees apart around the top plate of the cavity was φ degrees electrical. These were the results predicted for a rotating field and this therefore demonstrated experimentally that spinning fields are produced in the cavity. A small change in amplitude between probes can still arise because the energizing fields are not exactly equal or not exactly 90 degrees phase spaced as this superimposes a stationary field on top of the spinning one. Further details of the test are available and should anyone wish to set up a spinning field also included are some of the practical problems encountered and the solutions adopted.