VOLTAGE
3. WAVE PROPAGATION & MANIPULATION
3.5 Resonators
3.5.1 Overview
There are two ways of conceptualizing sample resonators for EMR when choosing a design for a specific application. For the most part, one can regard the sample resonator as a bandpass filter (Matthaei, Young, &
Malherbe, 1979), but one can find equally useful designs among the antenna literature (Aharoni, 1946; Kraus, 1950; Silver, 1949; Watson, 1947). The
1. MrcaowxVE ENGINEERING FUNDAMENTALS AND SPECTROMETER DESIGN 39 principal concern, however, when adopting filter or resonator structures for EMR sample resonators is that one must exercise care for ensuring that the magnetic field lines of the resonator mode is perpendicular to the DC field (i.e .flo .1HI because the resonance signal intensity goes as the sine of the angle between HI and flo(Bleaney& Stevens, 1953).
Like all forms of spectroscopy, the desired EMR sample resonator design maximizes signal-to-noise and sensitivity. The nature of the experiment dictates that the sample interact with an oscillatory high frequency magnetic field HI> and therefore one desires that the resonator structure maximally concentrates HJ within the sample volume. For example, in a rectangular cavity resonating in the TEI02 mode, the HI field line runs in a double loop with a concentration at the center, which is well suited to an axial sample in a tube. The so-called filling factor is a measure of the HI concentration within the sample volume and assumes a numerical value between zero and one (see Poole, 1983, pI56).
The other important sample resonator design factor is the quality, or Q, factor. In general , one wants a high Qfor highest sensitivity, and this is preferred for continuous wave experiments, but with high Q comes a propensity towards 'ringing'," which leads to long deadtime in pulsed applications or experiments that record a transient response. A Mims transmission line resonator for pulsed EMR typically features a Q of approximately 100, whereas a cylindrical TEall resonator for ENDOR might have a Q of 5000. Poole (1983, p124ff) outlines much of the relevant theory of Q, but, in general, Q is affected by surface finish and material of the resonator, the match of the resonator dimensions to the wave length, and the coupling arrangement to the transmission line.
3.5.2 Cavity Resonators
In Section 3.2 it was stated that an inductive iris can be fabricated by inserting thin strips through the wall of rectangular guide. A common form of filter can be designed by inserting a second inductor pair in series along the transmission line, and this filter can be 'tuned' to a specific frequency by adjusting the distance between these two strips so that t
=
I.. /2. One can introduce capacitance by closing off the aperture from the E-plane and forming an iris. This type of cavity/filter operates as a selective resonator because its geometry dictates what TE mode it will support. Inother words, a standing wave cannot form within the confines delineated by the irides unless the field lines 'fit' (boundary conditions for the electric and magnetic fields met). Cavity operating modes refer to the number of half-cycles that span the enclosure.20 A time-dependent, damped, oscillation of the standing wave amplitude following a rapid change in the wave amplitude.
A cavity resonator is a section of hollow waveguide that is closed off so that a standing wave is set up and confined to the enclosure. A simple model is a section of shorted waveguide section, which acts as a perfect reflector (see Section 3.1) that is delineated by a thin diaphragm. An aperture in the thin diaphragm permits entry of the traveling wave, and the wave reflected at the shorted end sets up a standing wave. If the diaphragm is located at a nodal point in the standing wave, the structure traps the energy and is said to resonate.
3.5.3 Stripline& Loop-Gap Resonators
Like the hollow waveguide, a pair of discontinuities in a microstrip transmission line forms a filter (Matthaei, Young & Malherbe, 1979), and again the dimension of the partitioned section is t... /2. With rectangular strips the HI field extends to either side and is a maximum along the short symmetry axis (cf, Edwards, 1981). The sample volume therefore has to be situated close to the strip and near the center, which makes these devices ideally suited for EMR studies of dielectric films (Wallace& Silsbee, 1991).
Another prospective application is spectroelectrochemistry; a t... /2 strip can be simultaneously operated as an electrode for standard electrosynthesis procedures because any small diameter conductor used to hook up the strip to the DC waveform generator will not support microwave transmission.
The so-called Mims (1965) cavity for pulsed EMR epitomizes the adaptation of stripline filter design for spectroscopy. A pair of waveguide sections run side-by-side and are terminated by a short. A rectangular t... /2 strip is situated approximately t... /2 away from the waveguide terminus and electrically couples to the standing waves in each of the parallel sections.
The strip is coupled and matched to the waveguide sections only to the extent afforded by tapers in the waveguide and small indentations to the either side of the strip; there is no iris or coupling structure that would maximizeQ.The sample sits in wells located to either side of the strip at the centerline.
The rectangular strip or linear conductor is a low inductance lumped parameter circuit element (see Terman, 1934; Young, 1977; Abrie, 1985) that will resonate when its length equals 2nt.../4 (open) or (2n+1)t... /4 (shunted), and the Qis commensurate with the strip's width. The H-field of the resonant TE mode resembles a half-wave along the longitudinal axis, and therefore the sample should be small and lie close to the strip's surface in order to maintain HI field homogeneity. One can, however, fold the rectangular strip around the short axis without drastically altering the resonant frequency, and this will improve the field homogeneity in the region that is now enclosed by the strip's inner surface. The correlation between the resonant frequency, as determined byt... /2, and the actual length
1. MICROWAVE ENGINEERING FUNDAMENTALS AND SPECTROMETER DESIGN 41
of the strip depends on the fringing capacitance and, as the radius of the folded strip decreases, the proximity of the two edges introduces a capacitance that will affect the equivalent RLC circuit and the resonant frequency of the device.
A folded rectangular strip is the most basic form of the so-called loop-gap resonator, which is a generic term that is applied to lumped-parameter sample resonators (cf Fronciz& Hyde, 1982). Loop gap resonators combine any number of loops (inductors) with gaps (capacitors) as part of an RLC resonant network, and their principle advantage for EMR applications comes from their compact size. Rather than a large rectangular cavity and its disperse field lines, the loop gap offers a small structure whose geometry (i.e .the circular 'loop') lends itself well to sample tubes and concentrates the H-field precisely at the sample; there is hence an improvement in the filling factor of the loaded resonator.
3.5.4 Helical Resonators
Helical resonators (Webb, 1962; Poole, 1967, p562ff, 752ft) are attractive for EMR applications that require broadband operation (e.g. swept-frequency zero field EMR, Bramley & Strach, 1983) or low Q(e.g. pulsed EMR), but tend not to be widely used because there is a perception that they are not competitive with conventional resonant cavities. There is also the matter of radiating mode as it relates to the helices' diameter; helices large in comparison to A(in other words, those suited to accommodate standard EMR sample tubes) radiate in the wrong mode for EMR applications.
One records EMR spectra as a change in the power level (i.e. the impedance) at the receiver as a material in a resonant circuit undergoes a change in permittivity. With resonant cavities the change in power is proportional to Q, that is dP
=
Y211'X"'P'Q, and therefore high-Q resonators are preferred. By comparison, however, the analogous equation (Siegman, 1960) for the change in power in helices is dP=
2rc·X"·P·S·N. In both equations P is the microwave power and X" is the complex susceptibility.The parameter 11 is the filling factor, S is defined as the slowing factor, and N is the number of carrier frequency wavelengths spanned by the helix. The slowing factor depends on the bandwidth of the helix, but may be nearly 1000 for a device of a few percent bandwidth, and the N-factor is approximately YJ at 10 GHz (Siegman, 1959; 1960).Itfollows that the helix is competitive with many cavities and lumped parameter devices such as the loop-gap resonator.
Siegman's analysis applies to small diameter helices, which radiate in the normal, as opposed to the axial, mode (see Kraus, 1950). The normal radiating mode is achieved with helices wound so that the diameter is approximately 0.1Aand an interwinding spacing of 0.05 A. The current
distribution in these helices tends to be sinusoidal if the circumference c is less than21J3,but this is rendered uniform as c - ').. (Marsh, 1951). One must also use care in selecting the conductor material since its diameter affects normal mode radiation (Tice & Kraus, 1949). A large helix that radiates in the axial mode is typically 0.32').. in diameter and has an interwinding spacing of 0.25 A..In the axial mode, the diameter of the conductor has little effect on the helix characteristics.