Progress in Nuclear Magnetic Resonance Spectroscopy 83 (2014) 1–20

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Progress in Nuclear Magnetic Resonance Spectroscopy journal homepage: www.elsevier.com/locate/pnmrs

Cavity- and waveguide-resonators in electron paramagnetic resonance, nuclear magnetic resonance, and magnetic resonance imaging Andrew Webb ⇑ C.J. Gorter Center for High Field MRI, Department of Radiology, Leiden University Medical Center, Leiden, The Netherlands

Edited by David Gadian and David Neuhaus

a r t i c l e

i n f o

Article history: Received 23 July 2014 Accepted 30 September 2014

Keywords: Cavity Waveguide Re-entrant cavity Traveling wave Dielectric resonator

a b s t r a c t Cavity resonators are widely used in electron paramagnetic resonance, very high field magnetic resonance microimaging and also in high field human imaging. The basic principles and designs of different forms of cavity resonators including rectangular, cylindrical, re-entrant, cavity magnetrons, toroidal cavities and dielectric resonators are reviewed. Applications in EPR and MRI are summarized, and finally the topic of traveling wave MRI using the magnet bore as a waveguide is discussed. Ó 2014 Elsevier B.V. All rights reserved.

Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Cavity resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Rectangular cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Cylindrical cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3. Re-entrant cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4. Cavity magnetrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5. Toroidal cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.6. Dielectric resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7. Circuit analogues of cavity resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.8. Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Cavity resonators in EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1. Rectangular and circular cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2. Coupling to cavity resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3. The axial uniform cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4. Cavities based on high permittivity materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5. Re-entrant cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.6. Loop gap (cavity) resonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Cavity resonators in NMR and MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1. Toroidal cavity resonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2. Magnetron-based cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3. Re-entrant cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.4. High permittivity dielectric resonators for MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

⇑ Address: C.J. Gorter Center for High Field MRI, Department of Radiology, C3-Q, Leiden University Medical Center, Albinusdreef 2, Leiden 2333 ZA, The Netherlands. Tel.: +31 71 526 5484; fax: +31 71 524 8256. E-mail address: [email protected] http://dx.doi.org/10.1016/j.pnmrs.2014.09.003 0079-6565/Ó 2014 Elsevier B.V. All rights reserved.

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5.

6.

Waveguides for MRI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Traveling wave MRI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Strategies for improving the sensitivity of traveling wave MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction A cavity can be defined as an enclosure formed by a surrounding three-dimensional conducting surface. Either a single or multiple small apertures are present through which electromagnetic (EM) energy can enter and exit. The dimensions of the cavity, which may be hollow or filled with a material with specified relative permittivity, are designed for a particular frequency relevant to the application, for example 2.45 GHz for a microwave oven. Cavity resonators are used in a number of different applications including linear accelerators, magnetron microwave sources, radar and in high power filter design. Relevant to the topic of this review article, cavity resonators are widely used for electron paramagnetic resonance (EPR), but are of increasing interest for nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). Although the dimensions of cavity resonators are intrinsically larger at the much lower (with respect to EPR) NMR and MRI frequencies, the steady increase in static magnetic field strengths has brought Larmor frequencies into a range where cavity resonator design becomes practical. Historically, it is interesting to note that the NMR probe used in the very first experiments reported in 1946 was a cavity resonator [1], shown in Fig. 1. As described by Pound [2] ‘‘As an open-ended coaxial resonator it would be a quarter wavelength long, 2.5 m, but a disk insulated from the lid by a thin sheet of mica as a loading capacitance shortened it to about 10 cm. The space below the disk that should contain circumferential rf magnetic flux at the cavity resonance was filled with about two pounds of paraffin wax, chosen because of its high concentration of hydrogen and its negligible dielectric loss.’’ In this review article, the basic principles of waveguides and cavities are introduced, followed by a description of their specific applications in EPR, NMR and MRI. 2. Cavity resonators

17 17 18 18 19

comparable to the wavelength of EM energy in free space. Although the cavity can have any three-dimensional geometry, for EPR, NMR and MRI experiments it is normally a rectangular cuboid (referred to as rectangular), cylindrical, or a re-entrant version of either of these two. The dimensions and co-ordinate geometries of rectangular and cylindrical cavities are shown in Fig. 2. A small gap, termed an aperture, is made in the structure so that EM energy can be fed into the cavity and the signal extracted. For simple analysis the walls of the cavity are assumed to be perfectly reflecting, resulting in regions of constructive and destructive interference within the cavity, with the locations dependent upon the frequency of the EM energy. Standing waves occur at certain frequencies, resulting in areas of very high and very low relative intensities, called modes, which arise from multiple reflections from the cavity walls. For the lowest frequency mode the dimensions of the cavity are approximately one-half wavelength, shown schematically in one dimension for clarity in a rectangular cavity in Fig. 3. Each standing wave mode has a particular geometric distribution of energy, which occurs at a certain frequency for given dimensions of the cavity. The energy in the standing wave alternates in time between electric and magnetic fields which are 90° out-ofphase. For large cavities there are a large number of different transverse electric (TE) and transverse magnetic (TM) modes. TE modes are defined as having electric field components only in the transverse direction, ie there is no longitudinal electric field component (Ez = 0), and a non-vanishing longitudinal magnetic field

a

2r

d

L

A cavity resonator is a very simple structure formed by an enclosure surrounded by conducting walls, with dimensions

b x φ

r z

z

y Fig. 1. A 30-MHz resonant cavity filled with paraffin as the proton sample for the first NMR experiments [1]. It is held at the Smithsonian Museum and has been cut open to reveal its inner structure. Figure reproduced with permission from [2].

Fig. 2. Basic geometries of (left) a hollow rectangular cavity and (right) a hollow cylindrical cavity. The top figures show the dimensions of the cavities, and the bottom the corresponding Cartesian (rectangular) and polar (cylindrical) co-ordinate systems.

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field component (Ez – 0). Each TE or TM mode is characterized by three subscripts (m, n, and p) which are covered in the next section. Detailed mathematical analyses can be found in many textbooks [3]. 2.1. Rectangular cavities For a rectangular cavity the following equation describes the resonant frequencies, f0, of both the TEmnp and TMmnp modes:

1 f 0 ¼ pffiffiffiffiffiffiffiffiffi 2 lr er

Fig. 3. (top) An EM wave entering a rectangular cavity through a small aperture is reflected from the walls of the cavity. Depending upon the relationship between the wavelength and the dimensions of the cavity, different modes of constructive and destructive interference can occur. (middle) In the case that the z-dimension is onehalf the wavelength, then a standing wave occurs due to reflections from the axial walls of the resonator giving rise to constructive interference with the incoming EM wave. (bottom) The resulting standing wave (shown in one-dimension only) has a high intensity at the centre of the cavity, with low intensities at either end.

component (Bz – 0). In TM modes there are magnetic field components only in the transverse direction, ie no longitudinal magnetic field component (Bz = 0), and a non-vanishing longitudinal electric

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 n2 p2 c m2 n2 p2 þ þ ¼ þ þ a2 b2 d2 2 a2 b2 d2

ð1:1Þ

with m, n and p the mode numbers, and a, b, and d the dimensions as shown in Fig. 2; lr and er are the relative permeability and permittivity, respectively, of the medium within the cavity. The three different subscripts, m, n, and p represent the number of periodic half-waves in the x-, y-, and z-directions, respectively. The electric field components are given by:

mp    x sin nbp y nap  pp  Ex  sin b y sin d z     Ey  sin map x sin pdp z

Ez  sin

ð1:2Þ

where m, n, p = 0, 1, 2. . .. The combination of m = n = 0 is not possible for TM modes and therefore the TM110 mode is the lowest frequency mode. The electric and magnetic field distributions of the TM110 mode of a rectangular cavity are shown in Fig. 4. The desirable characteristics of a cavity for EPR, NMR and MRI experiments are that the magnetic field has a high intensity where the sample

TM110

H-field

E-field

Fig. 4. Plots of the magnetic (top) and electric (bottom) fields for the TM110 mode of a rectangular cavity.

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TE101

H-field

E-field

Fig. 5. Plots of the magnetic (top) and electric (bottom) fields for the TE101 mode of a rectangular cavity.

is located, and the electric field has a low intensity. A high magnetic field gives efficient excitation and high signal-to-noise ratio (SNR), whereas a high electric field results in high power deposition and sample heating if the sample is conductive:

1 P ¼ rjEj2 2

ð1:3Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 X mn pp f 0;TM ¼ þ 2; 2p l e r L sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 2 1 X mn pp þ 2 f 0;TE ¼ pffiffiffiffiffiffi 2p le r L 1 pffiffiffiffiffiffi

ð1:5Þ

with m, n = 0, 1, 2. . .. and p = 1, 2. . .. (note that p cannot be zero). The lowest frequency TE mode is the TE101 mode, followed by the TE011 mode. The electric and magnetic field distributions for the TE101 mode of a rectangular cavity are shown in Fig. 5. From Figs. 4 and 5 it can be seen that the central part of the cavities contain a high electric field and a low magnetic field, exactly the opposite to optimum criteria outlined previously, and therefore these modes are not used for EPR or NMR experiments. As one considers higher order modes of increasing resonance frequency the first geometry with suitable characteristics is the TE102 mode, shown in Fig. 6: the sample is placed in the centre of the cavity.

defined in terms of Bessel functions Jm(Xmn) = 0 and J0 m(X0 mn) = 0. Values to calculate some of the lower frequency modes are given in Table 1. In cylindrical coordinates, m is the number of full-wavelength variations in the azimuthal (u) direction, and therefore the E and B fields are proportional to cos (mu) or sin (mu). The value of n represents the number of zeroes of the longitudinal field components (Ez, Bz) in the radial direction, and p is the number of halfwavelength variations in the longitudinal direction. Figs. 6 and 7 show magnetic and electric field distributions for the two most commonly used cylindrical cavity modes in EPR and NMR experiments, namely the TE011 and TM110 modes, both of which have a strong magnetic field and a weak electric field in the centre of the cavity. If the two modes in Figs. 7 and 8 are compared, advantages of each can be seen. For example the axial uniformity in the TM110 mode is far greater than in the TE011 mode. However, there is a much stronger magnetic field at the walls in the TM110 mode, which means that there are larger surface losses due to higher surface currents in the conducting walls.

2.2. Cylindrical cavities

2.3. Re-entrant cavities

In a cylindrical cavity the resonance frequencies of TE and TM modes are given by separate formulae:

A variation on the basic rectangular and cylindrical cavity geometries is a re-entrant cavity, in which the metallic boundaries

where P is the power absorbed, r the conductivity and E the electric field. For the TE modes the following equations hold:

Ez ¼ 0     Ex  sin nbp y sin pdp z mp    Ey  sin a x sin pdp z

ð1:4Þ

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TE102

H-field

E-field

Fig. 6. Plots of the magnetic (top) and electric (bottom) fields for the TE102 mode for a rectangular cavity resonator.

Table 1 Values of Bessel functions of zeroes. J0 m(x) = 0

m/n

Jm(x) = 0 n=1

n=2

n=3

n=1

n=2

n=3

m=1 m=2 m=3

2.405 3.832 5.136

5.520 7.016 8.417

8.653 10.173 11.620

3.817 1.841 4.301

7.016 5.331 8.015

10.173 8.536 11.346

of the conventional cavity extend back into the interior of the cavity. Fig. 9 shows an example of the concept, depicting a re-entrant version of the cylindrical cavity. Sections of the top and bottom of the cylinder extend back into the cylinder itself. The inductance of a re-entrant cavity is lower than the geometry from which it is derived due to the reduced surface area of the resonator, which means that a higher resonant frequency can be attained for a given overall physical size. Resistive losses are also lower, again due to the reduced surface area. The re-entrant design can also be used to concentrate either the electric or magnetic fields into specific parts of the resonator, as illustrated in Fig. 9(b) and (c). Some examples of different geometries of reentrant cavities are shown in Fig. 10. Examples of re-entrant cavities used in EPR and MRI are described in Sections 3.5 and 4.3, respectively.

2.4. Cavity magnetrons A cavity magnetron consists of a number of resonant cavities which are effectively joined in-series in a circumferential pattern, and produces high energy microwaves. A heated negatively-charged

cathode in the centre of the magnetron provides a source of electrons as shown in Fig. 11. A permanent magnet produces a focused magnetic field in the gap between the anode and cathode, and this field is focused by using high permeability pole caps (not shown). The electrons accelerate towards the positively-charged anode, with the exact path dependent upon their ejection energy and velocity. The electrons circulate around the different resonant cavities producing a high intensity microwave signal, which is transduced to the outside via an inductive pickup loop. Cavity magnetrons are commonly used in microwave ovens and in various radar applications. The resonant (operating) frequency of the device is determined by the exact size and geometry of the structure. The capacitance is provided by the gaps in the anode, and the inductance by the cylindrical holes in the anode, and the resonant frequency is given by the standard equation for an LC circuit:

f0 ¼

1 pffiffiffiffiffiffi 2p LC

ð1:6Þ

The cavity magnetron forms the basis for a few coil designs for MRI which are covered in Section 4.2.

2.5. Toroidal cavities Another important cavity design is a toroid, consisting of a number of turns of wire wound around (usually) a solid former, as shown in Fig. 12. In its lowest mode, the so-called ‘‘cyclotron mode’’, the magnetic field is completely confined within the toroid, with excellent separation between the magnetic and electric field components. A relatively uniform magnetic field is produced within the toroid if the ratio of the inner to outer diameter is close

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TE011 H field

E field

Fig. 7. Plots of the magnetic (top) and electric (bottom) fields for the TE011 mode in a cylindrical cavity resonator.

TM110 H field

E field

Fig. 8. Plots of the magnetic (top) and electric (bottom) fields for the TM110 mode in a cylindrical cavity resonator.

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(a)

(b)

(c)

Fig. 9. (a) Schematic of a re-entrant cylindrical cavity. (b) Electric and (c) magnetic field distributions inside the cavity.

(a)

(b)

(c)

(d)

Fig. 10. Different geometries of re-entrant cavities (right) derived from the ‘‘full’’ geometries (left). (a) Rectangular, (b) cylindrical, (c) disc-shaped, and (d) annular.

B field

Fig. 11. Schematic of an eight gap cavity magnetron used for producing high energy microwave radiation.

to unity, ie a relatively thin toroid is used. The value of the magnetic field is given by:



lr l0 NI 2pr

Fig. 12. Schematic of a wire-wound toroid, which produces a B field which is confined to the inside of the torus.

ð1:7Þ

where lr and l0 are the relative permeability and permeability of free space, respectively, N is the number of wire turns, I is the current through the wire, and r is the radius of the middle of the torus. Variations on the toroidal cavity used for NMR and MRI are covered in more detail in Section 4.1.

2.6. Dielectric resonators Another method of confining the electromagnetic energy within a structure is to construct it from materials with a very high permittivity [4], forming a dielectric resonator. In contrast to metallic cavity resonators, a fraction of the magnetic and electric fields of

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Fig. 13. Magnetic and electric field distributions in the four lowest frequency modes of a cylindrical dielectric resonator.

jω L

(a)

R

1/jωC

(b)

Fig. 14. (a) Lumped-element circuit model of a conventional RF coil. (b) Schematic showing the transformation from a lumped element resonant LC circuit to a cavity by rotating around the horizontal axis.

dielectric resonators does penetrate into the surrounding space. This field, however, is rapidly damped with increasing distance from the surface of the dielectric. The field distributions of different modes in such a dielectric resonator are very similar to those in hollow metal resonators of the same geometry if high permittivity materials are used: the lowest frequency modes are shown in Fig. 13. The sample is normally placed in the centre of the resonator inside a cylindrical hole which has been bored through the material. As can be seen from Fig. 13, for both the TE01 and HEM11 modes, this central location corresponds to a maximum in the magnetic field and minimum in the electric field, meaning that these modes are ideal in terms of high transmit efficiency and low sample heating. In contrast, the TM01 mode has maximum electric and minimum magnetic field at the centre of the cylindrical resonator. Therefore, if this mode were to be used, then the sample would be placed off-centre. For a cylindrical dielectric resonator a number of empirical expressions have been derived for the lowest resonant frequency modes:

f TE01d ¼ 2:921

 a 2

ce0:465 a r 0:691 þ 0:319  0:035 h h 2p a

ð1:8Þ

a2

ce0:436 a r 0:543 þ 0:589  0:05 h h 2pa

ð1:9Þ

f HEM11d ¼ 2:735

 

 ce0:468 a er  10 f TM01 ¼ 2:933 r 1  0:075  0:05 h 28 2p a a  a2

 0:071  1:048 þ 0:377 h h f HEM12 ¼ 23:11

a  a  cer 94 0:04  03:41e2:62a=h þ 1:55log 2pa h h

ð1:10Þ

ð1:11Þ

where a is the radius and h the height of the cavity resonator. 2.7. Circuit analogues of cavity resonators The classic parallel resonant circuit which forms the basis for all of the conventional RF coil designs (eg loop gap resonator (LGR),

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solenoid, saddle coil, birdcage resonator) is shown in Fig. 14. In a lumped element circuit there are resistive losses in the inductor due its finite conductivity, and these losses increase with frequency due to the decrease in skin depth. Capacitive losses also increase with frequency, as represented by the equivalent series resistance. Usually the inductor losses dominate, and so the circuit model has a resistor in series with the inductor, as shown in Fig. 14(a). Well-known expressions for the resonance frequency and Qfactor are:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R2 x0 ¼  LC L2 Q¼

ð1:12Þ

x0 L

ð1:13Þ

R

A schematic transformation of a parallel LC circuit to a cavity is also shown in Fig. 14. By schematically rotating the two-dimensional physical representation of a parallel-plate capacitor connected by two inductive strips into a three-dimensional structure, the inductance is decreased substantially meaning that the cavity has an intrinsically high resonant frequency. There are three basic loss mechanisms in a cavity: (i) resistive loss in the conducting walls of the structure, (ii) resistive losses in any dielectric which is used to fill the cavity (or in the case of a dielectric resonator the losses of the high permittivity material itself), and (iii) radiation loss through the aperture in the cavity. Usually the first mechanism is the dominant one, and the losses can be reduced by, for example, silver plating the conductor to prevent oxidation. As mentioned previously, different modes have different degrees of loss associated with them: the larger the surface currents associated with the particular mode the larger the losses. Thus, in general, TE modes have lower losses than TM modes due to lower surface currents.

waveguides are used to transfer microwave energy from the amplifier to the resonator. Analogous to the modes described previously for cavity resonators, i.e. shorted hollow waveguides, TE and TM modes of wave propagation can exist in a waveguide. When an EM wave propagates down a hollow structure, either the electric or magnetic field is transverse to the wave’s direction of travel, with the other field ‘‘looping’’ longitudinally with respect to the direction of travel, but remaining perpendicular to the other field. In TE modes the electric field is transverse to the propagation direction, whereas in TM modes it is the magnetic field which is transverse. As with cavities, different modes are distinguished by subscripts, although clearly only two are needed in contrast to the three for a closed cavity. Not all modes can propagate along a waveguide with given dimensions, with each mode having a specific cut-off frequency. Many textbooks provide excellent overviews of waveguides [5], the components which are relevant to magnetic resonance are summarized here. For a cylindrical hollow waveguide of diameter D the cut-off frequencies for TEmn and TMmn are given by: hollow

fc

¼

Waveguides are typically hollow structures with conductive walls, which are used to transfer electromagnetic energy from one region to another. For example, in EPR experiments rectangular

E-field TE11

cJ mn pD

ð1:14Þ

where Jmn is the nth root of the Bessel function of order m for the TM mode or the nth root of the differential coefficient of Jm for the TE mode. The TE11 mode has the lowest cutoff frequency of all the TE or TM modes that can propagate in a circular waveguide: the next lowest frequency mode is the TM01. The cut-off frequency for the TE11 mode is given by:

f c;TE11 ¼

1:841 pffiffiffiffiffiffi

pd le

ð1:15Þ

The cut-off frequency for the TM01 mode is given by:

f c;TM01 ¼ 2.8. Waveguides

9

2:405 pffiffiffiffiffiffi

pd le

ð1:16Þ

The relevance of waveguides to MRI is that at very high frequency the bore of a human-sized magnet forms a waveguide which can support propagating TE and TM modes [6]. This ‘‘traveling wave’’ concept is expanded on in Section 5. Fig. 15 shows the

B-field TE11

Fig. 15. Magnetic and electric field distributions for the lowest frequency TE11 waveguide mode.

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A. Webb / Progress in Nuclear Magnetic Resonance Spectroscopy 83 (2014) 1–20

fields associated with TE11 mode for a circular waveguide. The electric and magnetic fields effectively alternate in time with respect to both having their maximum value in the centre of the waveguide. In the TM01 mode the electric field alternates between having its maximum at the centre and at the edges of the bore, but there is zero magnetic field at the centre and the purely linear mode, ie it cannot be driven in quadrature, has only radial components and so is very inefficient for MRI.

example, has a value of 100 ns at X band. By increasing the bandwidth of the resonator the dead time can be reduced. Cavities can also be used for double-resonance experiments such as ENDOR (Electron-Nuclear DOuble Resonance) which uses both the EPR frequency and a much lower frequency for the NMR signal. ENDOR can also be performed in either CW or pulsed mode. In terms of the characteristics of the resonator, the measured voltage is proportional to the product of the filling factor (g) and the loaded Q factor, ie with the sample in place:

3. Cavity resonators in EPR

V / gQ L

Cavity resonators have found the greatest use in EPR studies. EPR hardware spans a wide range of frequencies, the most important frequency bands being L-band (1–2 GHz), S-band (2–4 GHz), X-band (8–10 GHz), Q-band (35 GHz) and W-band (90 GHz). Very high frequency EPR can be performed at frequencies up to 235 GHz with ‘‘semi-commercial’’ equipment and even higher with purely custom-built hardware. In general, loop gap resonators (see Section 3.6) are the preferred resonator geometry for experiments at frequencies below X band since cavity resonators are too large at these lower frequencies. At X-band either LGRs or cavity resonators can be used. Cavity resonators are generally preferred at Q- and Wbands since the LGRs become very small and difficult to fabricate. Above 100 GHz, the cavities themselves become too small and Fabry–Perot resonators are the probes of choice. It should be emphasized that these are just general guidelines. For example, cavities typically have a higher sensitivity than Fabry–Perot resonators, and so if one can construct a very small cavity resonator then this optimizes the SNR, as demonstrated by the Leiden group at 275 GHz [7]. EPR spectra can be recorded either in continuous wave (CW) or pulsed mode. CW EPR uses a constant frequency RF source and sweeps the external magnetic field from values slightly below the resonant condition for the different peaks to a value slightly higher. The RF energy source is typically a klystron and the energy is transferred to the cavity through a waveguide by means of an aperture termed an ‘‘iris’’. Amplitude modulation of the magnetic field with a frequency of typically 100 kHz increases the SNR by allowing phase sensitive detection, and so the probe must either contain modulation coils for producing 100 kHz, or else be transparent to external modulation coils. The main method of achieving the latter RF transparency is to make thin slots in the cavity. In pulsed EPR the spectrum is recorded by exciting a large frequency range with a single high-power MW pulse at a constant magnetic field B0. The bandwidth of the resonator is much larger, ie the Q-value is much smaller, than that of a CW EPR cavity. During and immediately after the RF pulse the probe must ‘‘ringdown’’ and signal detection cannot begin. This ‘‘deadtime’’ of the spectrometer depends on the bandwidth of the resonator and, for

sample

sample

ð1:17Þ

In general the Q value of a cavity is much higher than that of a LGR, but conversely the filling factor of the LGR is much higher. 3.1. Rectangular and circular cavities At low microwave frequencies the size of cavity resonators becomes inconveniently large and requires very large amounts of sample for optimum sensitivity via a high filling factor. Smaller volumes are typically studied using LGRs, which also have a higher resonator efficiency, defined as the magnetic field produced per unit input power. At higher frequencies the performance of LGRs and cavity resonators becomes similar. For example, Sidabras et al. [8] compared the performance of a five-loop-four-gap LGR with that of a cylindrical TE011 mode cavity resonator at 94 GHz, with very similar values for the efficiencies of both types of resonator. The most commonly used cavity resonator for low- to mediumfrequency EPR is the rectangular TE102 cavity. The sample is placed in the centre of the cavity, corresponding to the region of maximum magnetic field and minimum electric field, as shown in Fig. 16 and earlier in Fig. 6. For aqueous samples at X-band (9.5 GHz) the conventional procedure is to place a flat-cell cuvette vertically in the centre of the cavity, corresponding to the position of highest magnetic field and minimum electric field. However, as shown by Mett and Hyde [9] placing the cuvette horizontally, with the sample placed only in the area or high magnetic field, also results in good EPR spectra. The authors showed that by placing multiple flat-cells with optimized dimensions and spacing within the TE102 cavity, an increase in SNR can be achieved compared to a single cuvette in a parallel orientation. From a geometric point-of-view it is more favourable to use a square hole for sample access, and in a later paper [10] the authors showed that an increase of a factor of 2.7 in SNR, with respect to the conventional flat geometry, could be obtained using a 27 cell sample assembly. The key element in their SNR analysis was to consider the different loss mechanisms in the parallel and perpendicular orientations. Type 1 loss is associated with the tangential electric field within the sample, type II loss with the perpendicular electric field, and type III loss with partial cancellation of the electric field at the end of the sample. All three mechanisms are present for the parallel orientation, but only type I loss for the perpendicular orientation. As the frequency increases to Q-band and W-band the cylindrical TE011 cavity becomes the preferred geometry since the TE102 mode becomes too large in size. With the sample also placed vertically in a hole through the centre of the cavity, Fig. 7 shows that the magnetic field is a maximum, and electric field a minimum, in this configuration. Photographs of the two types of cavity are shown in Fig. 17. 3.2. Coupling to cavity resonators

Fig. 16. Schematic of the magnetic fields, electric fields and sample placement in a rectangular TE102 cavity used for pulsed EPR.

An extensive review of coupling between the waveguide and the resonator has been published by Mett et al. [11]. Coupling is achieved using an ‘‘iris’’ which is essentially a variable diameter

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8.4 cm

4.2 cm

Rectangular TE104

Cylindrical TE011

Fig. 17. Photographs of commercial EPR cavity resonators operating at X-band (9.7 GHz). The rectangular TE104 cavity is essentially two TE102 cavities placed back to back.

slot/hole between the waveguide and cavity. Typically, a screw can be used to change the size of the iris, as shown in Fig. 18. The iris is either a rectangular slot or a circular hole. If one takes the case of a rectangular hole then the iris is equivalent to two parallel conducting strips shorted at each end, supporting a TEM mode transverse to the long-axis of the iris. There are two criteria which must be fulfilled for impedance matching: first, the equivalent resistance of the LGR as seen by the iris must be transformed into the waveguide characteristic impedance. The second criterion is that the iris reactance should cancel the residual reactance of the LGR. Detailed derivations for the conditions to satisfy these two criteria are given in Mett et al. [11].

Iris screw

Iris

waveguide

cavity

Fig. 18. Schematic of coupling of a waveguide (left) to a cavity (right) through a small aperture in the cavity, termed the iris. Variable impedance matching for different samples is accomplished by a screw.

(a)

(b)

(c)

d

L

L=D D

L=2D

Fig. 19. (a) Schematic of a cylindrical uniform resonant cavity. The shaded area represents a dielectric material with thickness d of k/4. (b) and (c) EM simulations show that the field is axially uniform irrespective of the length-to-diameter ratio. Included in the simulation is an RF waveguide feeding through an impedance-matched iris in the side of the cavity. Figures (b) and (c) reproduced with permission from [12].

(a) endplate

sample

(b)

(c)

iris aperture

sample sample

endplate Fig. 20. Three different configurations of dielectric resonators used for EPR. (a) An annular resonator with two conducting endplates [14], (b) two discs operating in the TE01d mode with the sample placed between the discs [15] and (c) a small dielectric resonator placed inside, and mutually coupled to, a larger conventional waveguide [16].

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A. Webb / Progress in Nuclear Magnetic Resonance Spectroscopy 83 (2014) 1–20

3.3. The axial uniform cavity

1-loop-2-gap

An interesting variation in basic cavity design is the axially uniform resonant cavity introduced by Mett et al. [12,13]. In this design two pieces of dielectric material are placed above and below the metallic cavity as shown for a cylindrical geometry in Fig. 19. For the usual cylindrical TE011 mode the magnetic field varies cosinusoidally in the axial direction as shown in Fig. 7. In the axially uniform resonant cavity the dielectric slab thickness represents one quarter wavelength, i.e.:

k0 d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 4 er  1

2-loop-1-gap

S

S

5-loop-4-gap ð1:18Þ

where k0 is the wavelength in free space, and er is the relative permittivity of the dielectric materials. As explained by Mett ‘‘the physical reason the fields are uniform along the z axis in region L is that resonance occurs exactly at the cut-off frequency, where the wavelength goes to infinity’’. Since the field is axially uniform then the cylindrical mode can be described as a TE010 mode which cannot occur for a simple cavity. The authors suggest a notation TE0nU for such a mode, where U stands for uniform. An identical approach can be used for rectangular cavities, for example in the TE102 mode. An extensive theoretical analysis of the uniform axial cavity is given in Mett et al. [12]. 3.4. Cavities based on high permittivity materials High permittivity dielectric resonators have been used in EPR in two different configurations, either as stand-alone resonators or as ‘‘flux-concentrators’’ placed within a larger conventional cavity, as shown in Fig. 20. Rosenbaum (11) was the first to use the configuration shown in Fig. 19(a) as a hollow dielectric rod with metallic endplates at X-band. Energy was coupled into the cavity using an X-band iris through one of the endplates. Since materials with very low permittivity were used (Teflon, quartz, ruby, er < 10) higher order TE modes were used so the resonator would be a reasonable physical size, ie a resonator operating in the TE01 mode would have been physically much too large. Increasing the permittivity of the resonator allows the use of the TE011 mode, as shown for annular resonators by Walsh and Rupp [17] and Dykstra and Markham [18]. Double-stacked configurations, as shown in Fig. 19(b) in which the TE01n modes are strongly coupled, have found use in specialized applications such as high pressure EPR using diamond anvil cells [15]. Since cavity resonators are relatively easy to couple into using standard waveguides, many authors have used a small dielectric resonator placed within a larger metallic waveguide to increase the filling factor for very small samples. For a cylindrical cavity, the TE011 mode of the cavity is typically coupled (via mutual inductance) to the TE01d mode of the high permittivity annulus, as shown in Fig. 19(c), with both the high permittivity annulus and the cavity resonating at the same frequency. Two different modes result from

3-loop-2-gap

S

S

Fig. 22. Four different configurations of loop gap resonator. In the 3-loop-2-gap and 5-loop-4-gap the centre loop is for the sample (S), and the outer loops are for flux return. An inductive loop couples to one of the outer loops. In the 2-loop-1-gap the small loop is for the sample, with the smaller size relative to the flux return loop increasing the B1 intensity in the sample.

the mutual coupling, with the useful mode corresponding to current distributions producing parallel magnetic fields in the cavity and high permittivity resonator, and the less efficient unused mode characterized by a current distribution which produces anti-parallel magnetic fields. As shown by Mett et al. [16], and many other authors in the field of MRI [19], the efficiency of such a coupled setup is essentially that of the isolated dielectric resonator alone, without the challenges of coupling to a very small structure. This principle has also been used with ferroelectric resonators made from single crystals of potassium tantalate, KTaO3, [20,21] as well as strontium and calcium titanates with a relative permittivity of 160 [22]. An extensive review of the use of dielectric resonators in EPR was published in 1999 [23]. 3.5. Re-entrant cavities For EPR frequencies below 10 GHz it is not feasible to produce a TE102 cavity due to its large physical size. Although LGRs are often used, as mentioned previously an alternative is to construct a reentrant cavity. C- and X-band resonators which can be used to investigate large volumes of aqueous samples have been constructed using re-entrant cavities [24–26]: representative schematics of rectangular and cylindrical re-entrant EPR cavities are shown in Fig. 21. 3.6. Loop gap (cavity) resonators

Fig. 21. Schematic diagrams of a rectangular (left) and cylindrical (right) re-entrant cavity, with cut-outs to illustrate the geometry more clearly.

Although the loop gap resonator is typically not considered as a ‘‘cavity’’, many authors do refer to it as a ‘‘loop gap cavity resonator’’, and the basic structures of multi-loop multi-gap resonators have similar characteristics to re-entrant cavities. Many variations of the LGR have been devised, as shown in Fig. 22, primarily by the group of Hyde [8,27–35] and several reviews on the topic cover the many subtle details of design and construction [36]. In the multiloop multi-gap designs there is a large degree of separation of the magnetic and electric fields, with the magnetic field being maximum in the loops, and the electric field a maximum in the gaps.

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4. Cavity resonators in NMR and MRI Lumped-element resonant circuits consisting of discrete conductor lengths and capacitors are the basis for the vast majority of NMR and MRI RF probes. For high resolution NMR the most common configuration consists of an inner saddle coil [37] and outer Alderman-Grant coil [38], both of which may be tuned to multiple frequencies [39]. In MR microimaging the solenoid is most commonly used geometry. In animal and human MRI the birdcage coil [40] or TEM resonator [41] (described later in Section 4.3) is the most commonly used transmit coil, with arrays of mutually decoupled surface coils [42] forming the receive elements. Although cavity resonators have not been used in high resolution NMR, they have found increasing use in MR microimaging as well as animal and human MRI, particularly at high field. A quick calculation shows that, even for the lowest frequency TE mode of either a rectangular or cylindrical hollow resonator, the dimensions would be much too large to be practical, and so many different forms of making the cavity electrically smaller have been devised. These include the addition of discrete capacitors to reduce the resonant frequency, conversion to re-entrant cavities, and the incorporation of high permittivity materials; each of these approaches is covered in the following sections. 4.1. Toroidal cavity resonators As outlined in Section 2.5 toroidal cavities have the advantage that all of the magnetic field is confined within the cavity. Although the basic geometry per se is not very practical for probe construction, a couple of papers have filled a hollow torus with different liquids and obtained high resolution spectra [43,44]. A series-wound toroid as shown in Fig. 22 would have too high an inductance to operate at modern-day NMR frequencies, and so some method of increasing the self-resonance frequency must be incorporated. The group of Woelk and Rathke have done this by developing a cavity resonator [45–50] in which all the loops are effectively connected in parallel with a common ground. These can then be represented simply as outer and inner conducting surfaces, as shown in Fig. 23. This type of cavity has found extensive use in high pressure NMR measurements, in which the fact that the magnetic field is confined within the toroidal cavity means that the cavity can be placed within a high pressure vessel, without the close proximity of the walls of the vessel affecting the NMR performance. In contrast to the idealized toroid shown in Fig. 12, the inner diameter of the toroidal cavity in Fig. 22 is much smaller than the outer diameter. This results in a strong B1 gradient, with the B1 field inversely proportional to the distance away from the

central conductor. This very sharp drop-off means that experiments which utilize large B1 gradients are particularly appropriate for this type of resonator, eg the measurement of diffusion coefficients very accurately, without the issues associated with large switched B0 gradients such as strong eddy currents being induced in the solid conductor of the cavity [49]. The inhomogeneous B1 distribution in the radial direction can also be used for microimaging using the rotating frame approach [51]. A few publications have demonstrated the use of a gapped toroidal cavity resonator operating in the cyclotron mode [52,53], as shown in Fig. 23, although this design can equally well be described as a dual stacked re-entrant cavity resonator. The drivepoint at the axial midpoint is fed by an inductive loop inserted between the two stacked cylinders. The coil works in the TEM00 or cyclotron-mode, which produces a B1 field in the circumferential direction similar to that of the simple toroid in Fig. 12. As mentioned by the authors [52,53], higher order modes can be suppressed if the dimensions are chosen such that these modes occur below their cut-off frequencies. The advantages of such a resonator are that the near-complete enclosure of the resonating volume minimizes radiative losses, and the fact that its current path is distributed over the entire cylindrical surface minimizes resistive losses. Butterworth et al. [53] filled the body of the toroid with titanium dioxide, which has a relative permittivity of 70 in order to reduce the size of the resonator. Capacitive tuning and balanced matching were used to resonate the coil at 175 MHz. The Q of the unloaded coil was 1700, both empty and filled with TiO2. With a human subject in the filled resonator, the Q value was reduced to 1000. Images of a human leg are shown in Fig. 24.

4.2. Magnetron-based cavities Cavities based upon the magnetron geometry described in Section 2.4 have been used both as surface coil elements and also as volume coils. The concept of a magnetron-based cavity as a surface coil was first introduced by Mansfield [54], with subsequent developments by Rodriguez et al. [55,56], as shown in Fig. 25(a). The latter group also produced a design with eight cavities in a single-layer circular arrangement [57], with the cavity tuned to 63.8 MHz (1.5 T) by 330 pF of capacitance, as shown schematically in Fig. 25(b). In reference [57] the authors reported a higher SNR at greater depths using the cavity magnetron surface coil compared to a standard surface coil with equivalent diameter. Fig. 26 shows images obtained with a magnetron surface cavity at 1.5 T. A body coil

B field

loop

sample

Fig. 23. (left) Schematic of a wire toroid with each loop of the toroid in series with the next. (right) Practical implementation of a toroid used for high pressure NMR measurements, with a central conductor.

Fig. 24. (left) Schematic of a gapped toroidal cavity resonator, designed to operate at 4.1 T (170 MHz). The structure had an inner radius of 10.5 cm, outer radius of 12.5 cm, and overall length 27.5 cm. An aperture of diameter 12.4 cm was cut to allow sample access. (right) An axial image of a human lower leg, 5 mm slice thickness, gradient echo sequence, data matrix 256  256 voxels over a field-ofview of 200 mm. Figure (b) reproduced with permission from [53].

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(a)

(b)

Fig. 25. (a) An eight-element petal surface coil resonator with the equivalent lumped element circuit below [55,56]. (b) An eight segment cavity magnetron resonator with tuning capacitor [57].

Fig. 26. High resolution brain images acquired at 1.5 T with a cavity magnetron receive-only surface coil. Figure reproduced with permission from [57].

show that good quality images can be produced from rodents at high field, as illustrated in Fig. 27(b). 4.3. Re-entrant cavities

Fig. 27. (a) Schematic of one end-ring segment of a high frequency cavity resonator (57) with conductor rods joining the two ends of the resonator. (b) Images of a rat acquired at 500 MHz using a similar design, termed a slotted cage resonator, with four conductor legs. Image reproduced with permission from [59].

was used for transmit with the magnetron coil placed at the back of the head as a receive-only coil. The dimensions of the coil were 10 cm radius, 8 cavities each of diameter 2 cm, and a copper width of 2.5 cm. The cavity magnetron principle can also be used to design a volume coil [58,59], with resonant cavity end rings connected by a number of equi-spaced solid conductor elements, as shown in Fig. 27(a). This type of design can also be used as a half-resonator with a ground plane, the mirror currents on the ground plane producing a homogeneous field [60]. In reference [59] the authors

The most commonly re-entrant cavity resonator used in MRI is the TEM resonator devised by Vaughan et al. [41,61], with subsequent adoption by many other groups [62–71]. As an alternative to the (non-cavity-based) birdcage resonator [40,72], the TEM resonator has been used for body coil designs at 3 and 4 T, as well as for head and knee coils at many different field strengths. The basic form of the re-entrant cavity is shown in Fig. 28(a). In practice, the largely inductive coaxial line (cavity) is shorted on both ends (as part of the shield), and is resonated with primarily-capacitive open-ended coaxial line elements, as shown in Fig. 28(b). Compared with a lumped element coil circuit enclosing a given volume, the inductance of the coaxial cavity containing the same volume is significantly lower, meaning that a higher self-resonance frequency can be achieved. Alternatively, a fraction of the re-entrant cavity can be retained, joined by simple strips of copper, segmented by a series capacitor, as shown in Fig. 28(c): a series of these re-entrant cavity resonators were designed and their performance compared by Beck et al. [73]. In a TEM with N coaxial elements there are N/2 + 1 fundamental resonant modes. As shown in Figs. 28(d) the two degenerate M = 1 modes represent the two uniform quadrature modes for MR operation (see Fig. 29). Other forms of cavity have been designed for operation at very high frequencies: one such example being a slotted coaxial cavity resonator which operates as a human head resonator at 400 MHz (9.4 T) [74].

A. Webb / Progress in Nuclear Magnetic Resonance Spectroscopy 83 (2014) 1–20

(a)

15

(b)

shield

(c)

(d)

m=1 Fig. 28. (a) A cylindrical re-entrant cavity forms the basis for the TEM resonator. Capacitance is formed by the overlap of the inner conductor with a dielectric material surrounding it. (b) The physical realization of the concept uses a series of equally-spaced co-axial lines, in which the capacitance can be changed by varying the distance between the two parts of the split inner conductor (expanded view). Each end of the coaxial cable is connected to the shield. (c) A realization in which a portion (Dx) of the reentrant part of the cavity is maintained and conductor strips split by a capacitor join the two re-entrant sections. (d) One of the two degenerate m = 1 modes with a sinusoidal current distribution producing a homogeneous B1 field (arrows) inside the coil.

Fig. 29. (a) and (b) Transmit and receive images acquired with a 4T TEM transmit/receive body coil only. (c) and (d) Images acquired with the 4T TEM body coil for homogeneous excitation together with phased arrays for local reception. Detailed imaging parameters can be found in (58). Figure reproduced with permission from [61].

4.4. High permittivity dielectric resonators for MRI Dielectric resonators can also be used for high field MRI, providing that the permittivity of the material is sufficiently high that the resonator is of a practical size. The general advantages of dielectric resonators over conventional lumped element designs are that

they do not need extensive capacitive segmentation (which is used to reduce effective conductor lengths to below k/20), they are highly robust, and they also have a very high quality factor. Dielectric resonators have been designed both for high field (>14 T) microimaging in vertical bore magnets, and also for in vivo imaging in human 7 T horizontal bore magnets. As with dielectric-based

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Table 2 Dimensions of dielectric resonators with different permittivities designed for operation between 300 and 800 MHz.

TE01 HEM11 TE01 HEM11 TE01 HEM11 TE01 HEM11 TE01 HEM11 TE01 HEM11

Frequency (MHz)

Permittivity

Radius (cm)

Height (cm)

800 800 600 600 500 500 300 300 300 300 300 300

100 100 200 200 300 300 100 100 80 80 60 60

2 2 2 2 2 2 6 6 6 6 6 6

2 3 1.75 2.9 1.7 2.9 3.9 6.8 5.7 8.5 10.2 12

EPR resonators the most common geometry is a disc, with a hole in the centre into which the sample is placed. Two modes have been used in MRI resonators, the TE01 and HEM11 modes, both of which have zero electric field and maximum magnetic field at the centre. The EM field distributions are not greatly perturbed by the process of forming a central hole and the resonant frequency increases by only a small fraction. The TE01 mode is a linear mode, which has the disadvantage of requiring that the long axis of the disc (and therefore the sample) be perpendicular to the static B0 magnetic

field. There are two frequency-degenerate HEM11 modes which can be excited in quadrature, with the long axis of the disc now parallel to B0, which is much more convenient for in vivo applications. For either mode fine-tuning of the resonator can be performed using either a co-axial or co-radial conducting shield. For small samples the permittivity of the material used to form the resonator must be very high. Using Eqs. (1.6) and (1.7) some idea of the dimensions and permittivities required is shown in Table 2, both for vertical bore magnets (500 MHz and above) and also large-bore horizontal magnets (300 MHz). High permittivity resonators have been designed for microimaging using materials with permittivities of 320 [75] and 150 [76], for a 14.1 T (600 MHz) vertical widebore (8.9 cm) magnet. More recently, a larger resonator operating in degenerate quadrature HEM11 modes using a material with permittivity 175 has been designed for imaging human digits at 7 T (300 MHz) on a whole body system [77]. It is also possible to construct resonators from materials with lower permittivities, including most simply of all, water or its deuterated form. Wen et al. [78] constructed a series of cylindrical and annular dielectric resonators that used either water or D2O as the dielectric for a whole-body 4 T (170.75 MHz) MRI system. Aussenhofer et al. used a quadrature HEM11 mode to image the wrist at 7 T (298.1 MHz), also with water as the dielectric [79]. In vivo images acquired at 7 T from both the ceramic- and water-based dielectric resonators are shown in Fig. 30.

Fig. 30. High permittivity dielectric resonators designed for operation as quadrature HEM11 mode resonators at 7 T (300 MHz). (a) Resonator for digit imaging: ceramic annulus height 6.3 cm, diameter 8.6 cm, central hole diameter 2.5 cm, outer shield diameter 14 cm, outer shield length 8 cm. The material is barium strontium titanate with a relative permittivity of 175. (b) Image acquired with a three-dimensional gradient echo sequence: TR/TE 10 ms/5 ms, tip angle 10°, field-of-view 50  100  25 mm, data acquisition matrix 332  340  50, spatial resolution 0.15  0.3  0.5 mm, total image acquisition time 86 s. (c) Resonator for wrist imaging using water or deuterium oxide, relative permittivity 78. Resonator length 15.2 cm, outer diameter 14 cm and inner diameter 10 cm. (d) Image of the wrist: multi slice turbo spin echo sequence, TR/TE 8419/ 35 ms, turbo spin echo factor 8, data matrix 332  216, slice thickness 2 mm, in-plane resolution 0.5  0.5 mm, 1 signal average, total imaging time 7 min 50 s.

Fig. 31. Simulation of the magnetic field within and surrounding two very thin high permittivity discs operating in the TE01d mode. The relative permittivity is 2500, with a 1 mm thickness and 10 cm diameter. Note that there is very little coupling between the resonators despite their close proximity.

A. Webb / Progress in Nuclear Magnetic Resonance Spectroscopy 83 (2014) 1–20

128 MHz (3 Tesla)

298 MHz (7 Tesla)

400 MHz (9.4 Tesla)

Fig. 32. Electromagnetic simulations of the propagation of a TE11 mode traveling wave down the bore of a 60 cm diameter magnet at different frequencies. The colour scale depicts the magnitude of the magnetic field. The wave is launched from a patch antenna shown at the top of the bore. On a 3T scanner there is almost no propagation since 128 MHz is below the cut-off frequency of propagation. Propagation efficiency increases with frequency.

Table 3 Cut-off frequencies for first two modes in different magnet diameters (60 cm whole body MRI, 40 cm head-only MRI, 9 cm horizontal-bore animal).

TE11 TM01

17

60 cm

40 cm

9 cm

307.1 401.5

460.7 602.3

2047.6 2676.9

Fig. 33. (a) Electromagnetic simulations of the Poynting vector plot of a wave propagating from a patch antenna placed at the entrance of the magnet bore (top) with a human subject in the magnet. The intensities are represented by the colour (red highest, blue lowest) and the direction of the Poynting vector by the arrow. (right) Low tip angle gradient echo images of a human subject from patch antenna excitation, reproduced with permission from [81].

this finding actually suggests an alternative method of large FOV imaging [6], in which energy is introduced into the patient via a remote RF antenna designed to produce a ‘‘traveling’’ wave which can propagate through the bore of the magnet, described below. 5.1. Traveling wave MRI

High permittivity materials can also be used as surface elements, either as individual cavity resonators or with multiple elements combined into an array. The EM field patterns are similar to those of a conventional surface coil, but again there is the advantage of not requiring series capacitive segmentation, as well as individual elements being intrinsically very highly electrically isolated from each other in a large array. An example of simulated electromagnetic fields from two cavity resonators placed very close together is shown in Fig. 31. The resonators are designed with a very low aspect ratio, ie the height-to-diameter ratio, so that most of the energy enters the sample rather than remaining within the resonator.

5. Waveguides for MRI Constructing large coils for human body imaging is very challenging at high magnetic fields of 7 T and above. Adapting the typical body transmit coil design from a 3T magnet has a number of serious problems. One is the highly variable impedance of the coil loaded with different sized subjects, which means that the coil must be matched on an individual basis. A more important issue is that the resonator couples strongly to the waveguide modes of the bore of the magnet, outlined in Section 2.8, meaning that energy can be transported far away from the coil [80]. However,

A simple antenna can be placed at the end of the magnet bore, with the EM energy traveling down the bore of the magnet, as shown in Fig. 32. This concept forms the basis of what has been termed ‘‘traveling-wave’’ MRI [6]. This approach has the advantage of being extremely simple to implement from a technological point-of-view, without requiring a very large RF coil to be constructed. Table 3 lists the cut-off frequencies for the two lowest frequency modes, TE11 and TM01, for three different magnet configurations: whole body human, head-only human, and very high field horizontal bore animal. The values shown in Table 3 show that the cut-off frequency for the TE11 mode in a 60 cm diameter whole-body 7 T magnet is just slightly above the Larmor frequency (298 MHz). In a human 9.4 T magnet, both the TE11 and TM01 propagating modes are present, as they would be also for a head-only 14 T magnet, with the TE11 mode alone being applicable for an 11.7 T head-only magnet. For animal work, no modes propagate given that the strongest current magnet field for a wide-bore horizontal magnet is 17 T (720 MHz). When a human subject is placed in the magnet bore, a number of effects occur. The first is that the cut-off frequency decreases due to the increase in the effective permittivity, as described by Eqs (1.17) and (1.18). The second is that the mode structure becomes much more complicated, and is no longer a pure mode. Fig. 33(a) plots the Poynting vector, S, which represents the magnitude and power flow into the body, and is defined as:

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A. Webb / Progress in Nuclear Magnetic Resonance Spectroscopy 83 (2014) 1–20

(a)

(b)

(c)

(d)

Fig. 34. Four methods of improving the efficiency of traveling wave excitation. (a) Adding an internal dielectric liner to the magnet bore, (b) using a coaxial cavity to shield areas of the body which are not being imaged, (c) using a coaxial waveguide to improve impedance matching to the body, and (d) using a coaxial waveguide, dielectric liner and multi-element transmit array in combination with one another.

S¼EB

ð1:19Þ

As the wave travels down the bore of the magnet and encounters the body, the Poynting vector rotates from being parallel to the bore to being perpendicular. As this perpendicular energy flow enters the body, it is attenuated by the conducting tissue, and therefore the intensity in the centre of the body is relatively low. A whole body image showing this effect is illustrated in Fig. 33(b). 5.2. Strategies for improving the sensitivity of traveling wave MRI There are many challenges with the basic traveling wave approach outlined above, which typically has poor sensitivity compared to local coil excitation (84). One problem is that abrupt changes in wave impedance between empty regions of the bore and the body cause large amounts of energy to be reflected. For example, the shoulders cause strong reflections which results in a longitudinal standing wave in the head. A second disadvantage specific to imaging organs such as the prostate and liver is that the wave is absorbed in the body as it propagates along the bore, meaning that there is relatively little energy in the wave by the time it reaches the imaging volume, as shown in Fig. 33(b). Several approaches have been developed to overcome at least partially these limitations. The first is to decrease the cut-off frequency of the magnet bore by increasing the effective permittivity using a dielectric liner (80) consisting of an array of water tubes running along the axis of the magnet, as shown in Fig. 34(a). The further the Larmor frequency is above the cut-off frequency, the more energy is transmitted along the length of the bore. The liner reduces attenuation of the traveling wave in the longitudinal direction [82,83] and directs the radial power flow towards the subject. Using this approach the authors found a higher SNR in the middle of the brain by a factor-of-eight compared to measurements without the dielectric liner. Addition of a high permittivity liner also allows other modes to propagate down the scanner bore. Using 32 water tubes, up to eight waveguide modes could be supported [84]. By designing an appropriate multi-mode antenna stub array, each of these modes were excited separately, and the phase and amplitude of each of these modes adjusted to homogenize the B1 field over the brain, albeit at the cost of a much reduced transmit efficiency. Other approaches to this problem have also been suggested including the addition of an extra waveguide at the end of the magnet [85] or using multiple patch antennas as a transmit array [86]. The impedance mismatch at the shoulders can be mitigated by putting material with a similar relative permittivity, 40, by the

side of the head on top of the shoulders. This moves the region of mismatch away from the head and improves the B1 homogeneity within the brain. Ideally, the mismatch could be further reduced by using a tapered permittivity approach, similar to impedance matching using tapered microstrips. One method to limit the absorption of energy in parts of the body which are not being imaged is to use a coaxial waveguide section placed between the antenna and the region of interest [87]. Using a simple aluminium-foil conductor, Andrechenko et al. were able to obtain significantly enhanced transmission efficiency in the prostate. The coaxial waveguide approach can also be used to improve the impedance matching, analogous to classical transmission line theory, and expressions can be derived in terms of the required length of the coaxial section: details can be found in reference [87]. This approach can then be used to improve brain imaging, as also covered in reference [84] and shown in Fig. 34(c). Of course, different approaches can be combined, as shown in Fig. 34(d) in which a multi-element transmit array is combined with a dielectric liner and coaxial waveguide. As can be seen from Table 3, at 9.4 T then the lowest TM mode, TM01, can also be excited: this has been shown using a monopole antenna [88]. Simple B1 shimming can be performed using the TE and TM modes, although performance is limited by the fact that the pure TM mode has a central null in the magnetic field. An alternative approach to performing traveling wave MRI, which is not limited by cut-off frequencies has been introduced by Alt et al. using a TEM wave as the excitation pulse [89]. Their setup was a coaxial waveguide with an interrupted inner conductor to concentrate the energy at the desired region for imaging. This mode of wave propagation can exist only where there are two conductors since a TEM mode cannot propagate in a hollow waveguide. For the TEM mode both Ez and Bz are zero, with the electric field vectors oriented radially between the conductors, while the magnetic field vectors have a circular orientation around the central longitudinal axis: the TEM mode has no lower cut-off frequency. The B1 vectors have a circular orientation around the central longitudinal axis, but the B+1 field is zero in the centre. In the presence of the body, the fields are disturbed and some transmit field is present at the centre, but this approach is currently not very efficient.

6. Discussion and conclusion This review article provides background on the design criteria and performance of cavity resonators for many different types of

A. Webb / Progress in Nuclear Magnetic Resonance Spectroscopy 83 (2014) 1–20

electron paramagnetic and nuclear magnetic resonance experiments. Since static magnetic fields are becoming increasingly strong, with superconducting/resistive hybrids planned for upwards of 30 T for NMR experiments and whole body magnets of 11.7 and 14.1 T in advanced planning stages, it seems likely that cavity resonators and variations thereof will be instrumental in constructing efficient resonators for the associated very high Larmor frequencies. As in the past with examples such as the loop gap resonator, high frequency developments in EPR can readily translate to their NMR and MRI counterparts. The combination of cavity resonators, high permittivity materials and transmit arrays offer a distinct opportunity for new resonator designs in the future.

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Glossary of abbreviations CW: continuous wave EM: electromagnetic EPR: electron paramagnetic resonance HEM: hybrid electromagnetic LGR: loop-gap resonator MRI: magnetic resonance imaging Q-factor: quality factor RF: radiofrequency SNR: signal-to-noise ratio TE: transverse electric TEM: transverse electromagnetic TM: transverse magnetic

Cavity- and waveguide-resonators in electron paramagnetic resonance, nuclear magnetic resonance, and magnetic resonance imaging.

Cavity resonators are widely used in electron paramagnetic resonance, very high field magnetic resonance microimaging and also in high field human ima...
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