RESULTADOS FINALES

1.600 1.598 N 1.596 X s 1.594 o g 1.592 1 .590 1.588 1.586 T---r - 0 .1 0.0 0.1 0.2 A/? corr / mm 0.5 0.4

Figure (4.4) Halfwidth o f the TMOl 1 mode at a normalised plug displacement

N X 2.580 2.575 2.570 2.555 2.560 2.555 T--- r T--- 1--- 1--- r 2.550 - 0.1 0.0 0.1 0.2 A / Z c o r r / mm 0.3 0.4

analysis as equation (29), and so we might expect that they w ill be just as accurate i f carefully applied. Great care is particularly necessary, i f an opening is to be

considered, in employing the correct waveguide mode eigenvalue % for use in

equation (54), since the wrong choice can lead to very different calculated resonance

frequency shifts. A further d ifficu lty can arise for degenerate cylindrical cavity

modes, which have angular dependence o f electromagnetic fields. The orientation o f the E and H fields may not be same as that predicted from the unperturbed cavity theory, in which case the predicted resonance frequency shifts may differ somewhat from the actual shifts It is d ifficu lt to imagine how such a problem could be completely avoided. The same problem may, o f course, arise in estimating the shifts in resonance frequency caused by deformations o f spherical cavity boundary walls as has previously been noted in this section and in section (4.6). However, it is still instructive to estimate the effect o f the opening on the resonance frequency o f the doubly-degenerate TM 110 cylinder mode.

The cut-off length Hqq is the same as that for the TMOlO and T M O ll modes, since we assume that it is the (cut-off) TMOl waveguide mode that dominates w ithin

the opening and so A K = nb^h^Q = ;z(6V2%oi), as before. This expression for AF is substituted into equation (30) to give

a^L (4.7.70)

since aP' f o r j .= 0 modes. Substituting J | ( =-0.40276 [8],

h= (0.475 ± 0.015) mm, a = (9.520 ± 0.001) mm, L = (19.956 ± 0.004) mm and

= 2.40483 into this expression gives a predicted fractional shift o f

(a /*//,°o ) = “ (19.0± 1.8) ppm. Thus, we would predict that the opening causes a

reduction in the resonance frequency by 19 ppm o f the unperturbed resonance frequency, a shift o f a sim ilar magnitude but opposite ‘direction’ to that calculated and measured fo r the TMOlO mode. The shifts for the TMOlO and T M O ll modes are positive because it is the electric field that dominates in the vicinity o f the perturbation (the opening), whereas for the T M l 10 mode, the magnetic field is stronger in the region o f the perturbation. Equation (30) gives the fractional resonance frequency shift for both the/? = +1 and p = - \ components o f the T M l 10

mode, since equation (30) is exactly the same whether p is positive or negative, and so we predict that the two components are shifted by exactly the same amount, correct to first order. The opening itself is not expected to make any contribution to the resonance halfwidth, as discussed earlier, and, as the two components w ill ‘move together’ , the measured halfwidth ought to be the same in the presence o f the opening as it is in absence o f the opening. Unfortunately, measurements on the T M l 10 mode were not taken before final assembly o f the cylindrical resonator, in order to verify the predicted resonance frequency shift, but it is likely to be accurate to about 1 ppm o f the unperturbed resonance frequency, i f the orientation o f the fields w ithin the perturbed cavity is similar to that expected in an unperturbed cavity.

The spherical resonator used in this work has a 1 mm diameter gas inlet hole

drilled through the w all at its north pole (polar angle 6 = 0) [36]. Equation (7) enables the prediction o f the first-order fractional shift in resonance frequency for any TM /«0 mode (i.e. the w = 0 component o f any IM ln mode) and equation (18) predicts the fractional shifts for the /» = ±1 components o f any TM1« mode. Using these two equations, we can estimate the fractional shifts in resonance frequency due to the presence o f the gas inlet hole for all three components o f the T M lw modes measured in this work (T M ll, T M l2 and T M l3) and also the m = 0 components o f the TM21 and TM31 modes measured in this work. For the purposes o f these calculations it is assumed that the orientations o f the E and H fields in the perturbed cavity are the same as we would expect in the corresponding unperturbed cavity [this assumption was used to derive equations (7) and (18)]. It is d ifficu lt to estimate the accuracy o f this assumption since detailed measurements o f the resonance frequency shift due to such perturbations in the sphere have not been reported in the literature, but we could expect that the calculated shifts w ill have the same order o f magnitude as the actual shifts. The additional volume AK, which arises due to the presence o f

the hole, w ill be given by equation (54) as AV = 7[[b^/2%), where b is the hole

radius and % is the eigenvalue o f the dominant (cut-off) waveguide mode that is excited w ithin the opening by the cavity fields. For the /w = 0 components it is most

like ly that X ~ Xq\-> was found for the cylindrical resonator, since the TM O l waveguide mode is that mode with the lowest cut-off frequency whose E and H

the m = 0 components [see figures (3.2) and (3.5)]. Thus, for the TM /«0 modes,

the best estimate o f AV is n{b^ j l x For a TM lw mode, the m = +1 and m = - \

components have sim ilar E and H fields to the w = 0 component but the fields are orthogonal to those o f the w = 0 component and mutually orthogonal to each other.

Therefore, for the m = ± \ components, the most likely value o f % w ill not be Xqm for

a hole at 0 - 0 , but is more likely to be the cylindrical waveguide TE 11 mode

eigenvalue, since the E and H fields o f the TE 11 mode are more like ly to couple efficiently w ith the E and H fields o f the /w = ±1 components o f a spherical cavity

TM1« mode in the vicin ity o f a hole at ^ = 0.

The calculation o f the unperturbed cavity volume Vq requires the spherical cavity radius a and for these calculations the evacuated cavity radius Qq, at temperature

T = 273 K, w ill be used. Equation (4.3.5) gives ao(273 K ) = (39.99178 ± 0.00048) x lO'^m and thus

= (4 /3 );ra ’ = (267.9173 ± 0.0057) x 10"^ m \ Using AK = ;r(iV 2 Z o i ) for the

TM/nO modes and A F = n [ h ^ l l x [ ^ for the T M ln ±1 modes, the first order

fractional shifts in resonance frequency for the sphere modes used in this work were calculated and are presented in table (4.5).

Table (4.5) 10^(/Y hoie/A l) for TM modes in the sphere

771 = 0 771 = ± 1 T M llm + 0 . 1 1 -0.27 TM12w + 0 . 0 2 - 0 . 2 1 TM13m + O.OO7 - 0 . 2 0 TM21/W + 0.34 TM31m + 0.67 123

The estimated accuracy o f the calculated fractional shifts in resonance frequency presented in table (4.5) is ±5%, provided the perturbed cavity fields in the region o f the opening approximate to those o f the corresponding unperturbed cavity in the same region. Even i f the accuracy is only ±30%, then it is only for the TM31 mode that the calculated fractional resonance frequency shift w ill be in error by more than 0.1 ppm o f the unperturbed resonance frequency, which is on the order o f the experimental uncertainty o f the measured resonance frequencies. The fractional resonance frequency shifts for TM1«0 modes w ith n > A w ill be smaller than 0.007 ppm, and those fo r TM lw ±1 modes w ith n > A w ill be -0.20 ppm, correct to 2 significant figures. As was predicted, and measured, for the cylinder modes, the presence o f the gas inlet opening in the wall o f the spherical resonator is not expected to make a significant contribution to the resonance halfwidth o f any mode.

The boundary shape perturbation formula o f equation (1) has been shown to be capable o f accurately predicting the fractional shifts in resonance frequency due to localised protrusions, indentations and openings in the resonator w all and it may be expected to predict the fractional shifts caused by any form o f such perturbations provided their dimensions and positions w ithin the boundary wall can be determined accurately. Although the experimental verification o f equation (1) has been lim ited to two resonant modes in one cylindrical resonator, the agreement between theory and experiment was shown to be so good, particularly for such a large perturbation, that it is expected that it w ill be the accurate characterisation o f deformations rather than the validity o f equation (1), and all those equations carefully derived from it, which w ill prove to be the lim iting factor in determining the effect o f boundary shape deformations on the resonance frequencies o f resonant cavity modes.

4.8 Coupling

The measured microwave resonance frequency f y and halfwidth gj\j , o f a resonant cavity mode, depend on the electrical admittances o f the microwave generator (the source) and the instrumentation to measure the output power (the load). An admittance Y is the reciprocal o f the corresponding impedance Z and is defined as

Y = — = G + iB (4.8.1)

where G is the conductance and B is the susceptance. I f the reduced admittance o f the source is Gs + iBs , and the reduced admittance o f the load is Gjr, + iBj^, where

‘reduced’ means that the admittance is expressed as a fraction o f the corresponding characteristic admittance, then the first order fractional shift in a resonance frequency is given by [11, 50] ext V Jn (Bs + B ,) 2 & . (4.8.2)

and the contribution to the resonance halfwidth is [11, 50]

&ext =

{g, + G , ) f ,

2 a „

(4.8.3)

where (2ext is the external quality factor, which arises from the energy losses in the resistive parts o f the source and load impedances.

I f the source and load are perfectly matched to the coaxial lines attached to the resonant cavity, then we can put Bs = B i = 1 and G s = G i = 1, and equations (2) and (3) become /« 1 20, (4.8.4) ext &cxt = A (4.8.5)

Therefore, i f / ^ = / A there is a reduction in the resonance frequencies which is equal to the corresponding contributions to the halfwidth.

The insertion loss / o f a given mode, which is a measure o f the decrease in power on transmission through the cavity resonator for that mode, is defined as [11]

I _{= lOlog 10' A ' = 1 0 1 o g ,o (l/^ c )} (4.8.6)

where is the input power, is the output power and = (-Pout/'^n) is called the circuit efficiency, for the mode. The circuit efficiency is related to the external quality factor o f the mode by [11]

where 2u is the unloaded quality factor o f the mode. This is the quality factor o f the mode fo r an isolated cavity, and is that due to all other losses except those which contribute to <3^^,. <2„ is related to the total or loaded quality factor o f the mode by

Qn \ Qu QexX /

Therefore the total measured halfwidth o f the mode gjq is given by

(4.8.8)

«N = U . + « . X , ) ; V ( 4 8 9 )

where is the halfwidth o f the unloaded cavity and is the contribution to the halfwidth due to the losses in the resistive parts o f the source and load impedances.

Although we were unable to calculate the fractional resonance frequency shift o f equation (4), for either o f the cavities used in this work, measurements, described in reference 51, on a brass cylindrical cavity, coupled by simple probe antennae and w ith total quality factors o f approximately 3000 for the modes o f operation (which is very sim ilar to the quality factors o f the modes o f the cylinder used in this work) suggest that the fractional shifts could be on the order o f 1 ppm or less i f the source and load are very accurately matched to the coaxial lines attached to the resonant cavity. A general procedure for measuring the insertion loss o f equation (6) is given in reference 52.

This work is concerned w ith the measurement o f relative, rather than absolute, speeds o f light and, therefore, we are more concerned w ith changes in the resonance frequency than the absolute resonance frequencies themselves. Therefore, even i f the (unknown) resonance frequency shifts due to these external effects is much larger than suspected, the accuracy o f the dielectric constant measurements presented in

this work should not be compromised i f which appears in equation (4) does not alter significantly w ith frequency and/or pressure. This important point is explained more fu lly in section (4.10).

In this work, the resonant modes were excited and detected by means o f short, straight, probe antennae which were formed by extending the central conductors o f coaxial microwave cables into the cavity space. The antennae were positioned so as to be aligned w ith the E field lines o f the modes which were to be excited and detected. Generally, antennae are placed in a region o f strong E field (high density o f E field lines) so as to strongly excite/detect the mode, although this w ill give rise to larger perturbations due to the presence o f the antennae themselves. There is, then, a compromise to be made between sensitivity and perturbation [11]. Alternative means o f coupling include inserting loops, formed from the inner conductors o f coaxial cables, into the cavity space to lin k the magnetic flu x o f the H

fields o f the resonant modes, or waveguides (commonly cylindrical) are inserted into the walls o f the resonator in positions such that the E and H fields o f the waveguide mode couple effectively with the E and H fields o f the resonant cavity mode [11, 18, 20]. Both o f these alternative methods are likely to give rise to larger perturbations than small probe antennae because the physical disturbance they cause to the cavity fields would be greater [11].

The microwave cable on the input side transmits power from the source to the resonator and the radiation from the input antenna excites the resonant mode w ithin the cavity. The output antenna is inserted so as to couple only loosely w ith the resonator mode so that the power flowing out into the output cable is a very small but constant fraction o f the power dissipated w ithin the cavity (principally in the w all o f the cavity) [11]. The loose coupling is realised by using only very short probe antennae. Longer antennae would give rise to ‘tighter’ coupling which would

increase the power transferred, and thus the circuit efficiency rjc, but at the expense

o f the external quality factor 2ext- That is, i f rjc in equation (7) is large then the ratio

(ôcxt/2«) lïiust be small and so must be small i f Qu has remained relatively constant. So tight coupling, such as would be produced by long antennae, would

give rise to a small and thus a large contribution to the halfw idth , for a given

mode. This would degrade the resonance and make the precise measurement o f

resonance frequency d ifficu lt. Therefore we use loose coupling given by short antennae, and compensate for the lower circuit efficiency by using higher input power and very sensitive detection and measurement equipment.

Measurements o f the resonance frequency and halfwidth o f modes TMOlO and T M O ll were taken, using the brass cylinder, under ambient conditions o f temperature and pressure, for different lengths o f probe antennae. Figure (4.5) shows the fractional difference, for each mode, between the resonance frequency for a given length o f antennae /, and that measured w ith antennae o f nom inally zero length (where the inner conductors are cut o ff flush w ith the ends o f the coaxial cables). Also shown in figure (4.5) are the calculated fractional shifts due to the boundary shape deformation caused by the presence o f the probe antennae, obtained using equation (4.7.29). The differences between the measured and calculated fractional shifts are mainly attributed to the effects o f the resistive parts o f the source and load

impedances. Figure (4.6) shows the corresponding fractional changes in the

measured halfwidths, and figure (4.7) shows the ratios o f the output power, measured at the diode, to the input power, from the microwave generator, fo r both modes (see schematic o f microwave electronics in chapter 6). This ratio approximates to

the circuit efficiency rfQ and the correlations between rjc^ for a given length o f antennae, and the fractional resonance frequency shifts and fractional changes in halfw idth are clearly seen. It was necessary to open the resonator, by unscrewing the

top plate, in order to shorten the antennae after each measurement o f a n d and

therefore the length and average diameter o f the cylinder were very slightly different

for each o f the measurements. This gave rise to shifts in and, perhaps, changes in

itself, and so the presented resonance frequency shift measurements should only be considered accurate to approximately ±100 ppm in f y since this is the estimated inaccuracy introduced.

The smaller halfwidths were measured w ith antennae o f nominally zero length and it was found, fo r such antennae, that the resonances o f the TMOlO, TMOl 1 and T M llO cylinder modes could still be excited and detected w ith sufficient power to enable precise measurement o f the resonance frequencies and halfwidths in the presence o f any background signals. Thus, potentially, the most precise