Dual-compensated antisymmetric composite refocusing pulses for NMR

Journal of Magnetic Resonance 225 (2012) 81–92

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Dual-compensated antisymmetric composite refocusing pulses for NMR Smita Odedra a, Michael J. Thrippleton b, Stephen Wimperis a,⇑ a b

School of Chemistry and WestCHEM, University of Glasgow, Glasgow G12 8QQ, United Kingdom Brain Research Imaging Centre, Division of Clinical Neurosciences, University of Edinburgh, Western General Hospital, Edinburgh EH4 2XU, United Kingdom

a r t i c l e

i n f o

Article history: Received 18 August 2012 Revised 29 September 2012 Available online 17 October 2012 Keywords: Composite pulses Symmetry B1 inhomogeneity Resonance offset Phase cycling Pulse imperfections Error compensation

a b s t r a c t Novel antisymmetric composite 180° pulses are designed for use in nuclear magnetic resonance (NMR) and verified experimentally. The pulses are simultaneously broadband with respect to both inhomogeneity of the radiofrequency (B1) field and resonance offset and, as a result of their antisymmetric phase schemes, can be used to form spin echoes without the introduction of a phase error. The new dual-compensated pulses are designed analytically, using symmetry arguments and a graphical interpretation of average Hamiltonian theory. Two families of composite refocusing pulses are presented, one (ASBO-9) consisting of sequences made up of 9 simple 180° pulses and one (ASBO-11) of sequences made up of 11 simple 180° pulses. There are an infinite number of composite pulses in each family owing to a free phase variable in the solution to the average Hamiltonian equations and this allows selection of individual composite pulses with particular properties. Finally, a comparison is made between composite pulses designed using average Hamiltonian theory and those proposed for use in quantum computing by NMR. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction In nuclear magnetic resonance (NMR), a composite pulse is a continuous sequence of radiofrequency pulses of varying rotation (or ‘‘flip’’) angle and phase which can be used in place of a single pulse [1,2]. The purpose behind this substitution is that the composite pulse will have different bandwidth properties with respect to experimental frequency parameters, for example the nutation frequency x1 or the resonance offset X. The individual pulses used in NMR are imperfect in several ways, two of the most unavoidable imperfections being the spatial inhomogeneity of the radiofrequency field strength B1, which leads to a distribution of nutation frequencies across the sample, and the finite magnitude of x1 with respect to typical resonance offsets in the spectrum. Therefore much of the thrust of composite pulse research has been to find sequences which, relative to a single pulse, are broadband with respect to the frequency parameters x1 = |cB1| or X. Such composite pulses can be said to be compensated for the relevant imperfections [1–18]. Broadband composite 180° pulses have always been of particular interest as both inversion of longitudinal (z) magnetization and refocusing of transverse (x, y) magnetization are very sensitive to the common pulse imperfections. However, the early enthusiasm for such pulses was rather diminished when it was realized that they often do not show an advantage over simple 180° pulses when used for refocusing of transverse magnetization, as is required ⇑ Corresponding author. E-mail address: [email protected] (S. Wimperis). 1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2012.10.003

when forming a conventional (or Hahn) spin echo. For example, Levitt’s original composite 180° pulse, 90°90°180°0°90°90° [3], achieves broadband inversion of z magnetization with respect to the radiofrequency field strength but, when used for refocusing of x, y magnetization, produces a spin-echo signal with a phase that is very sensitive to the precise value of the radiofrequency field [4]. Therefore, the problem with a composite 180° pulse such as 90°90°180°0°90°90° is that, although its overall flip angle is broadband with respect to the radiofrequency field strength, its overall rotation axis is strongly dependent upon this parameter. A similar problem occurs with composite pulses compensated for resonance offset: for example, the broadband composite 180° pulse 90°0°270°180°360°0° [7] performs well for inversion of z magnetization over a range of offsets but when used as a refocusing pulse yields a spin-echo signal with a phase that is strongly dependent on offset. As a result of these considerations, there was much early interest in the design of phase-distortionless or constant-rotation composite pulses [8,12–18]. These have both overall flip angles and rotation axes that are essentially constant over a certain range of either the radiofrequency field strength or the resonance offset. However, the problem with such pulses is that their bandwidths are not nearly large enough for the imperfections that are routinely encountered in practical NMR spectroscopy. It seems that there are no short phase-distortionless composite 180° pulses that can compensate for a B1 field that drops to less than half its nominal value across the volume of the sample, or for a resonance offset parameter, jX=xnom j ¼ jDB=Bnom j, of 1 or greater, yet both these situations 1 1 occur frequently in experimental practice. In addition, most

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composite 180° pulses designed to compensate for B1 inhomogeneity have a poor performance with respect to resonance offset, often much worse than that of a simple 180° pulse. So there will always be either parts of the sample or signals with offsets that are outside the usable bandwidths of even the best so-called phase-distortionless composite pulses and an unwanted phase shift in the echo signal will nonetheless result. The composite pulse 90°90°180°0°90°90° has a phase-shift scheme that is time symmetric with respect to the center of the sequence. It has long been known that an antisymmetric composite pulse exhibits the favorable property that the rotation axis lies in a fixed vertical plane of the rotating frame (e.g., the x,z plane if the central pulse has phase 0°), for all x1 values and both on and off resonance [14,15,17]. Recently, we have shown that if the pure spin-echo signal is selected using either phase cycling and/or magnetic field gradients, as it ideally should be, then an antisymmetric refocusing pulse will produce an echo signal with constant phase, for all values of x1 and X [19]. As with any other phase-distortionless composite pulse, the effective bandwidth of an antisymmetric broadband pulse may not be sufficient to cover the full range of the relevant instrumental imperfection but, unlike a symmetric or asymmetric composite 180° pulse, it will not introduce any phase distortion into the spin echo. The purpose of this paper is to present the design and evaluation of new antisymmetric broadband composite 180° pulses that are dual compensated, that is, compensated simultaneously for both resonance offset and inhomogeneity of the radiofrequency field. With appropriate selection of the pure spin-echo signal, these antisymmetric refocusing pulses will produce no phase error whatsoever [19]. We believe that our recent work has shown that there is a clear need for such refocusing sequences yet, to our knowledge, only one composite pulse of this type has been presented in the literature before, in an appendix to Ref. [14] without any verifying simulations or experiments. Interestingly, we will show here that this sequence belongs to one of the two families of dual-compensated 180° pulses that we will present. Our new dual-compensated pulses will be designed analytically, with the aid of a graphical interpretation of average Hamiltonian theory.

2. Theory

The operator V in Eq. (1) describes the deviation of the actual pulse Hamiltonian H from ideal or nominal behavior, as described by Hnom. In this work, we are interested in two possible forms of the perturbation V. Either (i) V arises from a deviation of the true nutation frequency x1 from its nominal value xnom , as caused for 1 example by a spatial inhomogeneity of the radiofrequency field across the sample:

V inh ¼ ðx1  xnom 1 ÞðIx cos /ðtÞ þ I y sin /ðtÞÞ

ð3Þ

or (ii) it has its origin in an offset X of the rotating frame frequency from precise resonance:

V off ¼ XIz

ð4Þ

The time evolution of the spin system during the pulse is described by the propagator U(t), formally written:

 Z t  0 UðtÞ ¼ Tb exp i Hðt 0 Þdt

ð5Þ

0

b is the Dyson time-ordering operator. Just as H can be sepwhere T arated into two parts, one ideal and one a perturbation, so too can U(t):

UðtÞ ¼ U nom ðtÞU V ðtÞ

ð6Þ

 Z t  0 U nom ðtÞ ¼ Tb exp i Hnom ðt0 Þdt

ð7Þ

0

 Z t  e ðt 0 Þdt 0 V U V ðtÞ ¼ Tb exp i

ð8Þ

0

e ðtÞ ¼ U nom ðtÞ1 VU nom ðtÞ V

ð9Þ

e ðtÞ is the full Hamiltonian in the interaction representation or V ‘‘toggling frame’’ imposed by Hnom(t). The propagator Unom(t) describes the evolution of the spin system under the effect of the nominal radiofrequency field alone. Using the fact that Hnom(t) is piecewise time-independent, Eq. (7) can be written:

  U nom ðtÞ ¼ exp ixnom I/ ðt  sn1      s1 Þ     1 nomn     exp ix1 I/2 s2 exp ixnom 1 I /1 s1

2.1. Average Hamiltonian theory The use of average Hamiltonian theory to design composite pulses in NMR is well developed and described in elegant detail in Refs. [8,13], which themselves draw upon earlier work on windowless decoupling sequences by Burum et al. [20]. Therefore, the following outline account will be kept as brief as possible. Written in the usual rotating reference frame, the Hamiltonian during the pulse, either simple or composite, can be considered as the sum of two parts, Hnom and V:

H ¼ Hnom þ V

ð1Þ

Hnom ¼ xnom 1 ðIx cos /ðtÞ þ I y sin /ðtÞÞ

ð2Þ

Hnom describes the interaction of the ideal or nominal radiofrequency field with a spin system with total angular momentum operator I. The radiofrequency field produces a nominal nutation frequency xnom and has phase /(t). This phase is constant during 1 a single pulse and so piecewise-constant during a composite pulse; if the composite pulse consists of N pulses then /(t) has the value /n during the nth pulse. The duration of the nth pulse is sn. The sequence is then completely specified by the N phases /n and the N pulse durations sn, with the overall duration T = s1 + s2 +    + sN.

s1 þ    þ sn1 6 t 6 s1 þ    þ sn

ð10Þ

I/n ¼ Ix cos /n þ Iy sin /n

ð11Þ

Note that Unom(t) consists of a series of n  1 fixed rotations with flip nom angles xnom 1 s1 , x1 s2 , etc., and phases in the x, y plane of /1, /2, etc., and a final variable rotation with phase /n. According to Eq. (9), the order in which these rotations act on V is antichronological. If the propagator UV(T) in Eq. (6) were approximately equal to the identity operator, that is UV(T)  1, over a certain range of an experimental frequency parameter such as x1 or X, then the full propagator for the sequence, U(T), would approximate to the ideal propagator Unom(T) over that same range. In this case, and if the pulse durations and phases had also been chosen such that:

U nom ðTÞIz U nom ðTÞ1 ¼ Iz

ð12Þ

then we would have derived a broadband composite 180° pulse. As described by Tycko and coworkers [8,13], a good way to achieve UV(T)  1 over a certain range of an experimental frequency parameter is to make a Magnus expansion of it:

U V ðTÞ ¼ expfiðV ð0Þ T þ V 1 T þ   Þg

ð13Þ

S. Odedra et al. / Journal of Magnetic Resonance 225 (2012) 81–92

V ð0Þ T ¼

Z

T

e ðtÞ dt V

ð14Þ

0

V ð1Þ T ¼ 

i 2

Z

T

dt 1

Z

0

t1

e ðt 1 Þ; V e ðt 2 Þ dt 2 ½ V

ð15Þ

0

The zeroth-order average Hamiltonian V(0) and the first-order Hamiltonian V(1) are the first two terms in a power series expansion of an exact average Hamiltonian V. The terms Vinh(0) and Voff(0) are lininh(1) ear in ðx1  xnom and Voff(1) are 1 Þ and X, respectively, while V quadratic, and so on. Thus, by choosing the pulse phases and durations in the composite sequence such that V(0)T = 0, while simultaneously satisfying Eq. (12), we will have designed a composite 180° pulse that is broadband to ‘‘zeroth-order’’ in the relevant experimental frequency parameter, either ðx1  xnom 1 Þ or X. If greater bandwidth is desired then this can be achieved by designing ‘‘first-order’’ broadband composite 180° pulses that satisfy V(0)T = V(1)T = 0. 2.2. Composite 180° pulses In this work we will consider composite 180° pulses made of up of an odd number N of simple nominal 180° pulses:

180 /1 180 /2 . . . 180 /N

ð16Þ

The advantages of constraining the sequence to have this form are (i) that it satisfies Eq. (12) for all possible pulse phases {/1, /2 . . . /N}, leaving us free to choose these solely on the basis that they satisfy V(0)T = 0 or V(0)T = V(1)T = 0, and (ii) that, as will be shown shortly, the exclusive use of 180° pulses makes finding solutions for V(0)T = 0 or V(0)T = V(1)T = 0 particularly simple and intuitive. For inhomogeneity of the radiofrequency field we find for the composite pulse in Eq. (16): N   X V inhð0Þ T ¼ x1  xnom s I/0j 1

ð17Þ

j¼1

V

inhð1Þ

 2 2 N1 N i x1  xnom s XX 1 T¼ ½I/0k ; I/0j  2 j¼1 k¼jþ1

ð18Þ

where s is the duration of a simple 180° pulse. Note the prime on the phases in Eqs. (17) and (18). This indicates that these are the pulse phases transformed by the move to the interaction representation brought about by the fixed 180° rotations in Unom(t). These toggling-frame phases are related to the pulse phases by [17]:

/0k ¼ ð1Þkþ1 /k þ

k1 X ð1Þjþ1 2/j

ð19Þ

j¼1

Note that Eq. (19) is an example of an involution or a self-inverse function. For offset, we find:

V off ð0Þ T ¼ 2

V off ð1Þ T ¼

p

N X X

xnom 1 

I/0j þ90 þðj1Þ180

ð20Þ

j¼1

X nom 1

2 X N I/0j

2 x j¼1  2 X N1 X N X  2i ½I/0k þ90 þðk1Þ180 ; I/0j þ90 þðj1Þ180  nom

x1

ð21Þ

j¼1 k¼jþ1

Note that the toggling-frame phases in Eqs. (20) and (21) are related to those in Eqs. (17) and (18) in a systematic fashion and we have emphasized this by writing all of them in terms of the primed phases of Eq. (19). The fearsome looking equations in Eqs. (17)– (21) are, in fact, straightforward to derive from consideration of

83

Vinh(0)s, Voff(0)s, and Voff(1)s for a simple 180° pulse and the way in which they are transformed by the fixed 180° rotations in Unom(t). 2.3. Graphical solution of the equations Vinh(0)T = 0 and Voff(0)T = 0 Inspection of Eq. (17) reveals that it is simply a summation of transverse angular momentum operators, which we can represent as vectors in an x, y plane. Thus if we choose N = 3 and /01 ¼ 120 , /02 ¼ 0 , /03 ¼ 240 then the three vectors will sum to zero as shown in Fig. 1a and we will have Vinh(0)T = 0. These three toggling-frame phases can be converted to actual pulse phases using Eq. (19) and we find that /1 = 120°, /2 = 240°, /3 = 120°. We are at liberty to subtract a constant 120° from the phase of all the pulses and if we do this we find /1 = 0°, /2 = 120°, /3 = 0° or the well known broadband (for B1 inhomogeneity) phasedistortionless composite 180° pulse 180°0°180°120°180°0°. This is a time-symmetric composite pulse (s1 = sN, s2 = sN1, etc. and /1 = /N, /2 = /N1, etc.) yet its toggling-frame phases are antisymmetric (/01 ¼ /00 þ /010 , /02 ¼ /00 þ /020    /0N1 ¼ /00  /020 , /0N ¼ /00 þ /010 with, in this case, /00 ¼ 0 ). As discussed in the Introduction, our aim in this paper is to derive novel antisymmetric composite pulses so that we can exploit the special properties that (i) such pulses have overall rotation axes constrained to lie in a fixed vertical plane of the rotating frame for all x1 and X, and (ii), as a result, such pulses can be used to form spin echoes completely free of any phase distortion as long as the true echo signal is selected by phase cycling and/or magnetic field gradients. For the composite pulse in Eq. (16), it is easy to see that a choice of symmetric toggling-frame phases will result in a sequence with antisymmetric actual pulse phases. Thus if we choose N = 5 and the time-symmetric phases /01 ¼ /05 ¼ cos1 ð0:25Þ, /02 ¼ /04 ¼  cos1 ð0:25Þ, /03 ¼ 0 (or, approximately, /01 ¼ /05 ¼ 104:5 , /02 ¼ /04 ¼ 255:5 , /03 ¼ 0 ) then the five vectors will sum to zero as shown in Fig. 1b. Converting to actual pulse phases using Eq. (19) and subtracting a constant phase of 57.9° to make /3 = 0°, we find /1 ¼ /5 ¼ 46:6 , /2 ¼ /4 ¼ 255:5 , /3 ¼ 0 or the broadband (for B1) phase-distortionless composite 180° pulse 180°46.6°180°255.5°180°0°180°104.5° 180°313.4°. The overall rotation axis of this pulse lies in the x, z plane of the rotating frame for all x1 and X values. The ability of this antisymmetric composite 180° pulse, called F1 in Ref. [17], to form spin echoes without phase distortion was recently demonstrated theoretically and experimentally in Ref. [19]. Note that if we choose the conjugate angle, cos1(0.25) = 255.5°, instead above then this is equivalent to using the mirror-image of the vector sequence in Fig. 1b and we would simply generate the time-reversed composite pulse 180°313.4°180°104.5°180°0°180°255.5°180°46.6°. We do not consider this a distinct sequence as it has the same performance with respect to B1 inhomogeneity as its ‘‘forward’’ counterpart, while its offset performance is simply reversed with respect to the sign of the resonance offset. Inspection of Eq. (20) reveals that it too is simply a summation of transverse angular momentum operators. The only difference in this offset case is that the relationship between the toggling-frame phases and the pulse phases is different from the inhomogeneity case: for offset there is an additional +90° phase shift of the toggling-frame phases for the odd-numbered pulses and an additional 90° phase shift of the toggling-frame phases for the evennumbered pulses. Regardless of this, we can still choose N = 3 and /01 þ 90 ¼ 120 , /02  90 ¼ 0 , /03 þ 90 ¼ 240 , the three vectors will sum to zero as shown in Fig. 1a, and we will have Voff(0)T = 0. Subtracting the additional phase shifts we find /01 ¼ 30 , /02 ¼ 90 , /03 ¼ 150 and then, using Eq. (19) and after subtraction of a constant 30°, the actual pulse phases /1 = 0°, /2 = 300°, /3 = 0°. This is the broadband (for resonance offset) phase-distortionless composite 180° pulse 180°0°180°300°180°0°

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S. Odedra et al. / Journal of Magnetic Resonance 225 (2012) 81–92

(a)

(b)

2

1

1 2

3

3 5 4

odd

even

(c) 3

2

1 5

(d)

4

1

2

3

4

5 9

8 7

(e)

6

3 5 4

1

or

2

8

6

+ 11 7

10

6 4

10

2

8 9

Fig. 1. Graphical solution of the equation Vinh(0)T = 0 or Voff(0)T = 0 for the form of composite 180° pulse given in Eq. (16). The contribution from each pulse is a transverse (x, y) angular momentum operator of fixed length in the toggling frame relevant to B1 inhomogeneity (Eq. (17)) or resonance offset (Eq. (20)). (a) Toggling-frame solution for three 180° pulses (N = 3); this corresponds to the symmetric composite pulse 180°0°180°120°180°0° for B1 inhomogeneity or 180°0°180°300°180°0° for resonance offset. (b) Timesymmetric toggling-frame solution for five 180° pulses (N = 5); this corresponds to the antisymmetric composite pulse 180°46.6°180°255.5°180°0°180°104.5°180°313.4° for B1 inhomogeneity or 180°46.6°180°75.5°180°0°180°284.5°180°313.4° for resonance offset. (c) Toggling-frame solution for five 180° pulses (N = 5) that ensures that Vinh(0)T = 0 and Voff(0)T = 0 simultaneously by having both the odd- and even-numbered vectors sum to zero separately; as the phase / can take any value, this corresponds to the family of dual-compensated composite 180° pulses presented by Tycko and Pines, including the asymmetric sequence 360°0°180°120°180°60°180°120°. (d) Time-symmetric togglingframe solution for nine 180° pulses (N = 9) that ensures that Vinh(0)T = 0 and Voff(0)T = 0 simultaneously; as the phase / can take any value, this corresponds to the ASBO-9 family of antisymmetric dual-compensated composite 180° pulses generated by Eq. (22). (e) Time-symmetric toggling-frame solution for eleven 180° pulses (N = 11) that ensures that Vinh(0)T = 0 and Voff(0)T = 0 simultaneously; as the phase / can take any value and the even-numbered vectors can follow two inequivalent pathways, this corresponds to the ASBO-11 family of antisymmetric dual-compensated composite 180° pulses generated by Eq. (23).

(or equivalently 180°0°180°60°180°0°). This is not a very well known sequence but is one of a general series of symmetric solutions for Voff(0)T = 0 given in Ref. [13]. Similarly, we can choose N = 5 and find an antisymmetric broadband composite 180° pulse for resonance offset using the toggling-frame phases shown in Fig. 1b. This yields the broadband (for resonance offset) phase-distortionless composite 180° pulse 180°46.6°180°75.5°180°0°180°284.5°180°313.4°. This appears to be a

novel composite pulse although, like the composite pulse 180°0°180°60°180°0°, it is of little interest as its resonance offset performance is inferior to antisymmetric composite 180° pulses that make use of phases solely of 0° and 180° [15]. (For the avoidance of confusion, we should note in passing that such phasealternating antisymmetric pulses are also symmetric as 0° = 0° and 180° = 180°, hence the reference to ‘‘symmetric’’ in the title of Ref. [15].)

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S. Odedra et al. / Journal of Magnetic Resonance 225 (2012) 81–92

Table 1 Broadband composite 180° pulses discussed in this work. The figures of merit for compensation for B1 inhomogeneity (NB1 ) and resonance offset (NX) are defined in the text; a larger number corresponds to a broader bandwidth. Designation

Sequence

/ [cos1(0.25)]

NB1

NX

ASBO-9(7A) ASBO-9(7B) ASBO-9(7C) ASBO-9(7D) ASBO-9(B1) ASBO-9(X) ASBO-11(B1) ASBO-11(X) Simple LF (Ref. [3]) TP (Ref. [12]) TPG (Ref. [14])

360°162.4°180°313.4°180°284.5°180°0°180°75.5°180°46.6°360°197.6° 180°46.6°360°255.5°180°75.5°180°0°180°284.5°360°104.5°180°313.4° 180°46.6°180°75.5°360°255.5°180°0°360°104.5°180°284.5°180°313.4° 180°104.5°180°29°180°104.5°540°0°180°255.5°180°331°180°255.5° 180°268.5°180°62°180°6.5°180°131°180°0°180°229°180°353.5°180°298°180°91.5° 180°176.5°180°173°180°320.5°180°288°180°0°180°72°180°39.5°180°187°180°183.5° 180°260°180°103.5°180°187°180°119.5°180°292°180°0°180°68°180°240.5°180°173°180°256.5°180°100° 180°220°180°287.5°180°235°180°31.5°180°68°180°0°180°292°180°328.5°180°125°180°72.5°180°140° 180°0° 90°90°180°0°90°90° 360°0°180°120°180°60°180°120° 180°256°180°52°180°0°180°128°180°0°180°232°180°0°180°308°180°104°

104.5° 255.5° 75.5° 180° 311° 108° 188° [104.5°] 52° [104.5°] – – – –

0.178 0.325 0.325 0.150 0.485 0.174 0.369 0.178 0.062 0.046 0.106 0.473

0.428 0.324 0.332 0.308 0.228 0.444 0.240 0.540 0.100 0.096 0.256 0.224

To derive a dual-compensated 180° pulse, we need to find a sequence that has Vinh(0)T = 0 and Voff(0)T = 0. As described by Tycko and coworkers using a theoretical framework similar to average Hamiltonian theory [12,14], this can be done by noting that for the odd-numbered pulses the toggling-frame phases for inhomogeneity and resonance offset differ only by a constant +90° and that for the even-number pulses the two toggling frames differ only by a constant 90°. Therefore, by ensuring that the angular momentum vectors for the odd-numbered pulses and for the evennumbered pulses separately sum to zero then a dual-compensated pulse should result. For example, with N = 5 and /01 ¼ 120 , /03 ¼ 0 , /05 ¼ 240 and /02 ¼ /, /04 ¼ / þ 180 then both the oddand even-numbered vectors will sum to zero as shown in Fig. 1c and we will have Vinh(0)T = Voff(0)T = 0. As noted by Tycko and coworkers, this generates an infinite family of composite 180° pulses as the phase / is a free variable [12,14]. These five toggling-frame phases can be converted to actual pulse phases using Eq. (19) and, after subtraction of 120°, we find that /1 = 0°, /2 = 120°  /, /3 = 120°  2/, /4 = 300°  3/, /5 = 4/. With / = 120° this gives the asymmetric dual-compensated 180° pulse 360°0°180°240°180°300°180°240° (or equivalently 360°0°180°120° 180°60°180°120°) and with / = 90° it gives the symmetric composite pulse 180°0°180°30°180°300°180°30°180°0° (or equivalently 180°0° 180°330°180°60°180°330°180°0°). To obtain an antisymmetric dual-compensated 180° pulse it is clear that we need to have at least N = 9 so that the toggling-frame phases for both the odd- and even-numbered pulses can be symmetric, as shown for N = 9 in Fig. 1d. Thus with /01 ¼ /09 ¼ 104:5 , /03 ¼ /07 ¼ 255:5 , /05 ¼ 0 and /02 ¼ /08 ¼ /, /04 ¼ /06 ¼ / þ 180 then an antisymmetric pulse with Vinh(0)T = Voff(0)T = 0 will result. Note that again there is a free phase variable /, thereby generating an infinite family of composite 180° pulses (which we will call ASBO-9; ASBO stands for AntiSymmetric, for B1 and Offset). These nine toggling-frame phases can be converted to actual pulse phases using Eq. (19) and, after subtraction of a constant phase to make /5 = 0°, we find that:

/1 ¼ /9 ¼ 4/ þ cos1 ð0:25Þ ¼ 4/ þ 104:5 /2 ¼ /8 ¼ 3/ þ 2 cos1 ð0:25Þ ¼ 3/ þ 209 /3 ¼ /7 ¼ 2/ þ cos1 ð0:25Þ ¼ 2/ þ 104:5

ð22Þ

/4 ¼ /6 ¼ / þ 180 /5 ¼ 0 For example, with / = 0° this gives the antisymmetric dual-compensated 180° pulse 180°104.5°180°209°180°104.5°180°180°180°0°180°180° 180°255.5°180°151°180°255.5°. Further antisymmetric dual-compensated pulses clearly exist with N = 11, N = 13, etc. Of these, however, only sequences with N = 11 can be derived in a closed form dependent on a single free

phase variable / as with N = 9. This can be seen in Fig. 1e, where the toggling-frame phases of the six odd-numbered pulses follow a time-symmetric pathway consisting of two equilateral triangles (to which we have added the free phase /) and the toggling-frame phases of the five even pulses follow the same symmetric pathway as was used in Fig. 1b and for the odd-numbered pulses in Fig. 1d. These toggling-frame phases correspond to the antisymmetric dual-compensated family of composite 180° pulses (which we will call ASBO-11):

/1 ¼ /11 ¼ 120  5/ /2 ¼ /10 ¼ 240  cos1 ð0:25Þ  4/ /3 ¼ /9 ¼ 240  2 cos1 ð0:25Þ  3/ /4 ¼ /8 ¼ 240  cos1 ð0:25Þ  2/ /5 ¼ /7 ¼ 120  /

ð23Þ

/6 ¼ 0 Note that, in this case, the two conjugate angles, cos1 (0.25) = 104.5° and cos1(0.25) = 255.5°, generate distinct composite pulses, with different bandwidths with respect to both B1 inhomogeneity and resonance offset. The two conjugate angles correspond to the two mirror-image vector pathways for the even-numbered toggling-frame phases shown in Fig. 1e and these have distinct relationships with the odd-numbered pulse vector pathway for all values of the relative phase /. 2.4. Higher-order terms As derived above, the N = 9 and N = 11 dual-compensated pulses in Eqs. (22) and (23) appear to be ‘‘zeroth-order’’ sequences with Vinh(0)T = Voff(0)T = 0. To investigate the first-order average Hamiltonians, we can use the equations for Vinh(1)T and Voff(1)T in Eqs. (18) and (21), respectively. Remembering that our antisymmetric sequences have symmetric toggling-frame phases, it can be seen by inspection of Eq. (18) that they will have Vinh(1)T = 0 as, for example, ½I/01 ; I/02  ¼ ½I/0N1 ; I/0N . Similarly, for Voff(1)T, the operator part of the first term on the right-hand side of Eq. (21) is identical to that in Eq. (17) and so will equal zero if Vinh(0)T = 0, while the second term on the right-hand side of Eq. (21) will be zero for a symmetric sequence of toggling-frame phases. Thus, the dualcompensated pulses in Eqs. (22) and (23) have Vinh(1)T = Voff(1)T = 0 on account of their symmetry properties. However, when B1 inhomogeneity and resonance offset are present simultaneously then there is a further first-order average Hamiltonian to consider. In this case, the perturbation V in Eq. (1) is given by:

V inhþoff ¼ V inh þ V off

ð24Þ

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S. Odedra et al. / Journal of Magnetic Resonance 225 (2012) 81–92

and we find to first-order that:

V

inhþoff ð1Þ

T¼V

inhð1Þ

T þV

off ð1Þ

T þV

inh;off ð1Þ

T

ð25Þ

with the inhomogeneity-offset cross term:

Z t1 Z

i T e inh ðt 1 Þ; V e off ðt 2 Þ dt 1 dt 2 ½ V 2 0 0  e off ðt 1 Þ; V e inh ðt 2 Þ þ ½V

V inh;off ð1Þ T ¼ 

ð26Þ

For the composite 180° pulse in Eq. (16) we then find:

V inh;off ð1Þ T ¼ 

N X 2ðx1  xnom 1 ÞX I/0j þ90 þðj1Þ180 2 ðxnom j¼1 1 Þ

þ

N1 X N pX

2

i

!

½I/0k ; I/0j þ90 þðj1Þ180  þ ½I/0k þ90 þðk1Þ180 ; I/0j 

j¼1 k¼jþ1

ð27Þ The first term in the large brackets on the right-hand side of Eq. (27) is identical to the operator term in Eq. (20) and so will equal zero if Voff(0)T = 0, while some thought shows that the second expression in the large brackets on the right-hand side (the sum of the two commutators) will be zero for a symmetric sequence of toggling-frame phases f/0i g. Thus, on account of their symmetry properties, the two families of dual-compensated pulses in Eqs. (22) and (23) are fully broadband up to first order, including the inhomogeneity-offset cross term: Vinh(0)T = Voff(0)T = Vinh(1)T = Voff(1)T = Vinh,off(1)T = 0. 2.5. Individual broadband solutions As a result of the free phase variable / in Eq. (22) it is possible to choose pulses in the ASBO-9 family that have pulse phases /1 = /2 (ASBO-9(7A)) or /2 = /3 (ASBO-9(7B)) or /3 = /4 (ASBO-9(7C)) or /4 = /5 = /6 (ASBO-9(7D)). These ‘‘7-pulse’’ solutions are given in Table 1. Of course, the duration of these sequences is still equal to 9 simple 180° pulses but it is possible that the reduced number of phase shifts might make them more convenient or robust in experimental practice. Similarly, there are ‘‘9-pulse’’ solutions for the ASBO-11 series of composite pulses but these would appear to be of lesser interest in view of the existence of the ASBO-9 family. The individual ASBO-9 or ASBO-11 sequences of most interest are those with the greatest bandwidths, either with respect to x1 = |cB1| or X. These were found by incrementing the free phase / in 1° steps from 0° to 360° in Eqs. (22) and (23), allowing for both cos1(0.25) = 104.5° and cos1(0.25) = 255.5° in Eq. (23), and simulating the range of B1 =Bnom and jX=xnom j ¼ jDB=Bnom j values 1 1 1 where the echo amplitude retains >99.0% of its full amplitude. For B1 inhomogeneity, this range can be expressed as the fraction, NB1 , of the range of B1 =Bnom from 0 to 2 where the echo amplitude 1 retains >99.0% of its amplitude at B1 ¼ Bnom . Thus NB1 can take 1 values from 0 (no broadband properties) up to 1 (a hypothetical, perfect broadband pulse for B1 inhomogeneity). For resonance offset, the performance of an antisymmetric pulse sequence is asymmetric with respect to X = 0 (it is only symmetric for a symmetric sequence). Therefore, we calculate the range of offset parameters X=xnom where the echo amplitude retains >99.0% of its amplitude 1 at X = 0 for both positive and negative offsets and then assign to the offset figure of merit, NX, the smaller of the two values. Thus NX can take values from 0 (no broadband properties) up to 1 (a hypothetical, perfect broadband pulse for resonance offset). A value of NX of, for example, 0.402 means that the spin-echo amplitude has >99% of its full amplitude between X=xnom ¼ 1 0:402 and þ 0:402 and that possibly, at one extreme of this range, the composite pulse might have an even broader bandwidth. Table 1 gives the optimum ASBO-9 and ASBO-11 pulses for B1 inhomogeneity (ASBO-9(B1) and ASBO-11(B1)) and resonance off-

set (ASBO-9(X) and ASBO-11(X)) found by this method, where their bandwidths are compared with the ‘‘7-pulse’’ ASBO-9 sequences mentioned above. It is important to stress that all members of the ASBO-9 and ASBO-11 families of composite pulse are dual-compensated for B1 inhomogeneity and offset and our computer search is merely for the purpose of identifying solutions with particular properties; indeed, the minimum values of NB1 and NX found in these families (NB1 ¼ 0:142 and NX = 0.224 for ASBO-9 and NB1 ¼ 0:118 and NX = 0.184 for ASBO-11) are still greatly in excess of the values for a simple 180° pulse. For comparison, Table 1 also lists some composite pulses from the literature, along with their NB1 and NX values, and further compares them with a simple 180° pulse. The sequence 90°90°180°0°90°90° (here designated LF) is the original broadband composite 180° pulse proposed by Levitt and Freeman [3]; it is not of the phase-distortionless type and symmetric and so its NB1 and NX bandwidth values are even lower than those of the simple 180° pulse. The sequence 360°0°180°120°180°60°180°120° (here designated TP), which has been discussed above in Section 2.4 [12,14], was designed to be phase-distortionless and dualcompensated with respect to both B1 inhomogeneity and offset X but because it is asymmetric and not antisymmetric its NB1 and NX values are significantly lower than, e.g., the ASBO-9(X) pulse. Finally, the 9-pulse antisymmetric composite 180° pulse 180°256°180°52°180°0°180°128°180°0°180°232°180°0°180°308°180°104° (designated TPG here) was presented in Ref. [14] without any verifying simulations or experiments yet can be seen to be similar to the sequence ASBO-9(B1) and to yield similar NB1 and NX values. Indeed, it corresponds to Eq. (22) with /  308°. The ‘‘fixed point’’ theory used in Ref. [14] to derive composite pulses is compared in that work with average Hamiltonian theory and the two are shown to have similarities. In this context, it is perhaps interesting to note that the sequence TPG appears to correspond to the solution to Eq. (22) with the smallest magnitude of Vinh(2)T. 2.6. Passband solutions Passband composite 180° pulses have inversion B1 profiles that are locally narrowband for B1  0 and locally broadband for B1  Bnom ; that is, they leave an initial z-magnetization state 1 unaffected over a range of low B1 values but act as conventional, compensated 180° pulses for B1 values near nominal. Therefore, they have the ideal B1 profile for spatial localization experiments [18,21–23]. For example, antisymmetric passband composite 180° pulses have recently been used in an improved version of the Cory–Ritchey ‘‘depth pulse’’ experiment for suppression of the unwanted ‘‘background’’ signal from the probe often found in 1 H MAS NMR experiments [24]. Passband pulses can be very sensitive to resonance offset, however, with only the symmetric 9-pulse sequence presented in Ref. [22] specifically designed to be dual-compensated for B1  Bnom . 1 For use in spin-echo experiments, it is worth examining if any of our dual-compensated antisymmetric families of broadband composite 180° pulses simultaneously show additional narrowband (i.e., passband overall) behavior. A narrowband pulse has Hnom = 0 in Eq. (1) and so there is no transformation into the toggling frame. For B1 inhomogeneity, the zeroth-order average Hamiltonian in the narrowband region is simply given by

V inhð0Þ narrow T ¼ x1 s

N X I /j

ð28Þ

j¼1

where the phases {/j} are the actual pulse phases rather than the toggling-frame phases in Eq. (17). For an antisymmetric sequence this is a purely real function and it is a simple matter to plot inhð0Þ V narrow T as a function of / in Eq. (22) or (23) and read off the values

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Table 2 Passband composite 180° pulses discussed in this work. The figures of merit for compensation for B1 inhomogeneity (NB1 ) and resonance offset (NX) are defined in the text; a larger number corresponds to a broader bandwidth. Designation

Sequence

/ [cos1(0.25)]

NB1

NX

ASBO-9(PB1) ASBO-9(PX) ASBO-11(PB1) ASBO-11(PX)

180°26.5° 180°149°180°64.5°180°160°180°0°180°200°180°295.5°180°211°180°333.5° 180°104.5° 180°119°180°284.5°180°270°180°0°180°90°180°75.5°180°241°180°255.5° 180°165° 180°27.5°180°130°180°81.5° 180°273° 180°0°180°87° 180°278.5°180°230°180°332.5°180°195° 180°190° 180°263.5°180°217°180°19.5° 180°62° 180°0°180°298° 180°340.5°180°143°180°96.5°180°170°

340° 90° 207° [104.5°] 58° [104.5°]

0.234 0.214 0.253 0.190

0.228 0.368 0.236 0.508

of / where V inhð0Þ narrow T ¼ 0, which will correspond to passband solutions. Note that because of the antisymmetry of the pulse phases inhð1Þ none of these sequences will have V narrow T ¼ 0. They will have off ð0Þ V narrow T ¼ 0 but this is not a desirable feature; ideally, one would wish for broadband behavior with respect to offset for B1  0 but this is not possible. For the ASBO-9 family generated by Eq. (22) there are 6 passband solutions between / = 0 and 360° and for the ASBO-11 family generated by Eq. (23) there are 6 solutions for cos1(0.25) = 104.5° and 8 for cos1(0.25) = 255.5°. The most promising of these sequences in terms of broadband B1 performance (ASBO-9(PB1) and ASBO-11(PB1)) and broadband offset performance (ASBO-9(PX) and ASBO-11(PX)) are given in Table 2. 3. Results and discussion 3.1. Simulations The performance of broadband composite 180° pulses in a spin-echo experiment was simulated assuming an initial state

rinitial = Iy (the result of a perfect 90°0° pulse on an Iz state). The pure echo signal was selected by incrementing the overall phase of the refocusing pulse through the four steps of the ‘‘Exorcycle’’ phase cycle [25,26] and summing the resulting magnetizations appropriately over the four steps such that the ideal result should be rfinal = +4Iy. We then plotted the desired in-phase magnetization component hIy i ¼ Trfrfinal Iy g=TrfI2y g and (where necessary) the unwanted out-of-phase component hIx i ¼ Trfrfinal Ix g=TrfI2x g as a function of B1 =Bnom (to study performance in the presence of B1 1 inhomogeneity) or DB=Bnom (to study performance in the presence 1 of a resonance offset), where DB = X/c is the residual static field in the rotating frame, with the resonance offset from the transmitter frequency X and the gyromagnetic ratio c. The simulations shown in Fig. 2a and b illustrate the effect of the symmetry of a composite 180° pulse on the phase of the refocused magnetization as a function of normalized B1 field strength, B1 =Bnom . In Fig. 2a the hIyi magnetization produced by a simple 180° 1 pulse, as well as the broadband composite pulses LF, TP and TPG, has been simulated. When used as a refocusing pulse, the response

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

LF TP TPG

-2

(a)

-3 0

0.5

1

1.5

2

0

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

(c)

-3 0

0.5

1

B1 / B1nom

1.5

2

(b)

-3 0.5

1

1.5

2

ASBO-9(7C) ASBO-9(B1) ASBO-11(B1)

(d)

-3 0

0.5

1

1.5

2

B1 / B1nom

Fig. 2. Simulations of the refocusing performance of broadband composite 180° pulses in a spin-echo experiment as a function of B1 =Bnom (to study performance in the 1 presence of B1 inhomogeneity). The simulations assume an initial state rinitial = Iy and a refocusing pulse that has been subjected to the four steps of the Exorcycle phase cycle. (a) In-phase magnetization component hIyi and (b) the unwanted out-of-phase component hIxi produced by a simple 180°0° pulse and the composite pulses LF, TP and TPG (see Table 1). In (b), only the simple refocusing pulse (which is technically antisymmetric) and the antisymmetric TPG sequence yield a zero out-of-phase component. (c) In-phase hIyi and (d) unwanted hIxi spin-echo components produced by three of the new antisymmetric ASBO sequences (see Table 1). In (d), all of these antisymmetric sequences yield a zero out-of-phase component.

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4

4

3

3

2

2

1

1

0

0

-1

-1

(a)

-2 -1

-0.5

0

0.5

1

-1

4

3

3

2

2

1

1

0

0

-1

-1

(c) -1

-0.5

0

0.5

(b)

-2

4

-2

LF TP TPG

-0.5

0

0.5

ASBO-9(7A) ASBO-9( ) ASBO-11( )

(d)

-2 1

B1 / B1nom

1

-1

-0.5

0

0.5

1

B1 / B1nom

Fig. 3. Simulations of the refocusing performance of broadband composite 180° pulses in a spin-echo experiment as a function of DB=Bnom (to study performance in the 1 presence of resonance offset). The simulations assume an initial state rinitial = Iy and a refocusing pulse that has been subjected to the four steps of the Exorcycle phase cycle. (a) In-phase magnetization component hIyi and (b) the unwanted out-of-phase component hIxi produced by a simple 180°0° pulse and the composite pulses LF, TP and TPG (see Table 1). In (b), only the simple refocusing pulse (which is technically antisymmetric) and the antisymmetric TPG sequence yield a zero out-of-phase component. (c) In-phase hIyi and (d) unwanted hIxi spin-echo components produced by three of the new antisymmetric ASBO sequences (see Table 1). In (d), all of these antisymmetric sequences yield a zero out-of-phase component.

of LF to B1 inhomogeneity is in fact narrowband compared with the simple 180° pulse. TPG shows the most broadband response of these composite pulses. It can be seen in Fig. 2b that the symmetric sequence LF and the asymmetric composite pulse TP both give rise to an unwanted Ix magnetization component in the presence of a B1 inhomogeneity. Conversely, the simple 180° pulse and the antisymmetric TPG pulse sequence do not yield any unwanted Ix magnetization when the B1 field strength deviates from the nominal value. Antisymmetric refocusing sequences are therefore essential, as they will produce a spin echo with perfect phase. Three of our novel antisymmetric dual-compensated composite pulses have been simulated in Fig. 2c and d as a function of B1 =Bnom : ASBO-9(7C), which shows the best response to B1 inhomo1 geneity amongst the ‘‘7-pulse’’ solutions to the ASBO-9 family; ASBO-9(B1), which has the greatest B1 bandwidth of all our new composite pulses and is slightly more broadband than the existing pulse sequence TPG; and ASBO-11(B1) – this sequence has the best performance in the presence of B1 inhomogeneity in the 11-pulse set. Owing to the antisymmetric phase schemes of these pulse sequences, no out-of-phase Ix magnetization is produced by any of these composite pulses. The performance of a simple 180° pulse and various composite refocusing pulses with respect to resonance offset is shown in Fig. 3. Fig. 3a and b shows simulations of a simple 180° pulse and the composite 180° pulses LF, TP and TPG from the literature. As with the case of B1 inhomogeneity, only the antisymmetric sequences do not produce any Ix magnetization off-resonance. The

composite pulses LF and TP will reintroduce phase distortions into the spectrum as out-of-phase Ix magnetization results when the transmitter is not on resonance. Fig. 3c and d shows the offset performance of three new dual-compensated 180° pulses: ASBO9(7A), the optimum ‘‘7 pulse’’ solution of the ASBO-9 set for compensating offset; ASBO-9(X), which performs very slightly better; and ASBO-11(X), which has the greatest bandwidth for resonance offset. These pulse sequences are all more broadband for offset than the dual-compensated pulses TP and TPG. The dual-compensated nature of the ASBO composite pulses can be fully appreciated in the contour plots shown in Fig. 4. The spinecho experiment was simulated using a simple 180° pulse (Fig. 4a) the composite refocusing pulses LF (Fig. 4b), TP (Fig. 4c), ASBO9(B1) (Fig. 4d), ASBO-9(X) (Fig. 4e) and ASBO-11(X) (Fig. 4f), and the resulting hIyi contours at 99%, 95% and 50% plotted as a function of B1 =Bnom and DB=Bnom . Note that the symmetric LF and asymmet1 1 ric TP sequences will also yield unwanted Ix magnetization, which we do not show here. In Fig. 5a the refocusing performance of the passband composite 180° pulses ASBO-11(PB1) and ASBO-11(PX) is shown as a function of B1 =Bnom . These sequences show broadband behavior near 1 the nominal field strength, but are locally narrowband at low B1 =Bnom values. ASBO-11(PX) has the greatest value of NX of these 1 sequences, indicating the best broadband offset performance. However, as shown in Fig. 5b where this pulse sequence has been simulated with DB=Bnom ¼ 0:1, it is not compensated for reso1 nance offset in the narrowband region ðB1 =Bnom 0:1Þ, as expected, 1

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B1 / B1nom

2.0

2.0

(a)

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0 1.0

0.5

0.0

0.5

1.0

(d)

0.5

0.0

0.5

1.0

1.0

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0 0.5

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

1.0

(f)

1.5

1.0

0.5

2.0

(e)

0.0

(c)

0.0 1.0

2.0

2.0

B1 / B1nom

2.0

(b)

0.0 1.0

0.5

0.0

0.5

1.0

1.0

0.5

B1 / B1nom

B1 / B1nom

B1 / B1nom

Fig. 4. Simulations of the in-phase hIyi component yielded by various broadband composite 180° pulses in a spin-echo experiment as a function of B1 =Bnom and DB=Bnom . The 1 1 results are presented as a two-dimensional contour plot, with contours drawn at 99%, 95% and 50% of the maximum. The simulations assume an initial state rinitial = Iy and a refocusing pulse that has been subjected to the four steps of the Exorcycle phase cycle. (a) Simple 180° pulse; (b) LF composite pulse; (c) TP composite pulse, (d) ASBO-9(B1) composite pulse; (e) ASBO-9(X) composite pulse; (f) ASBO-11(X) composite pulse. The LF and TP composite pulses are not antisymmetric and will also yield an unwanted out-of-phase hIxi component (not shown).

4

4

(a)

(b)

3

3

2

2

B / B1nom

180

1

ASBO-11(P B1) ASBO-11(PB

1)

ASBO-11(P R) ASBO-11(P

)

00

+0.1 +0.1

1

-0.1 0.1

0

0 0

0.5

1

1.5

2

B1 / B1nom

0

0.5

1

1.5

2

B1 / B1nom

Fig. 5. Simulations of the refocusing performance of passband ASBO pulses in a spin-echo experiment as a function of B1 =Bnom . The simulations assume an initial state 1 rinitial = Iy and a refocusing pulse that has been subjected to the four steps of the Exorcycle phase cycle. (a) In-phase magnetization component hIyi (the hIxi component is zero) produced by a simple 180°0° pulse and the passband composite pulses ASBO-11(PB1) and ASBO-11(PX) (see Table 2). In (b), the performance of the passband sequence ASBO-11(PX) is simulated on resonance and at resonance offset parameters of DB=Bnom ¼ 0:1. 1

despite being well compensated in the broadband region ðB1 =Bnom 1Þ. 1 3.2. Experiments Experiments were performed on a 500 MHz Bruker Avance III NMR spectrometer equipped with a standard bore B0 = 11.7 T magnet. The sample was H2O in D2O in a nonspinning 5-mm glass NMR tube in a conventional saddle-coil probe. The calibrated 90° pulse

length of 1H was 13.1 ls, corresponding to a nutation frequency, x1/2p = |cB1|/2p, of 19 kHz. 1H NMR spectra were acquired using a 90° – s – R – s – spin-echo pulse sequence, where s = 1 ms and R is the simple or composite refocusing pulse. A 16-step phase cycle was used, comprising a 4-step Exorcycle nested with a 4-step CYCLOPS [19]. The spin-echo spectra in Fig. 6a were acquired using either a simple 180° refocusing pulse, or the symmetric LF composite pulse, or the ASBO-9(B1) pulse, as shown, and with the transmitter

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S. Odedra et al. / Journal of Magnetic Resonance 225 (2012) 81–92

LF

ASBO-9(B1)

(a) nom 1

=

B1 / B1nom = 0

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.8

4.7

4.6

ppm

4.9

4.5 ppm

4.9

4.8

4.7

4.6

ppm

(b) nom 1

= 0.75

B1 / B1nom = 0

(c) nom 1

=

B1 / B1nom = 0.25

(d) nom 1

= 0.75

B1 / B1nom = 0.25

4.9

4.8

4.8

4.7

4.7

4.6

4.6

ppm

Fig. 6. 1H (500 MHz) NMR spin-echo spectra of H2O in D2O (2 ms total echo interval, 90° pulse duration of 13.1 ls, spin–lattice relaxation interval of 6 s, four-step Exorcycle phase cycle applied to the refocusing pulse). Results are shown using a simple 180° pulse, the LF composite 180° pulse and the ASBO-9(B1) composite 180° pulse (see Table 1). (a) Pulses with correctly calibrated durations and on-resonance. (b) Constituent pulses in composite refocusing sequences set to 75% of their correctly calibrated durations but still on-resonance. (c) Pulses with correctly calibrated durations but off-resonance with DB=Bnom ¼ 0:25. (d) Constituent pulses in composite refocusing sequences set to 1 75% of their correctly calibrated durations and off-resonance with DB=Bnom ¼ 0:25. 1

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S. Odedra et al. / Journal of Magnetic Resonance 225 (2012) 81–92

frequency on resonance with the single peak in the 1H spectrum. Using the calibrated B1 field strength, all pulse flip angles were set to their nominal values. The spectrum recorded with a simple 180° pulse was phased and this same phase correction was applied to other on-resonance experiments. A saddle-coil probe produces a B1 field that is significantly inhomogeneous and so phase distortions are present in the spectrum recorded with the LF refocusing pulse, as this symmetric sequence is generating out-of-phase magnetization. Note, however, that the antisymmetric ASBO-9(B1) sequence yields an in-phase signal with greater amplitude than the simple 180° pulse. To exacerbate the effect of B1 inhomogeneity intentionally, the spin-echo experiments were repeated with the constituent elements of the refocusing pulses misset to 75% of their correctly calibrated durations (e.g., a 90° pulse was replaced with a 67.5° pulse, a 180° pulse replaced with a 135° pulse, etc.). The results are shown in Fig. 6b. Again, the composite pulse LF produces a spectrum with a large phase error. The antisymmetric composite pulse ASBO-9(B1) yields a spectrum with perfect phase and a signal of higher amplitude than the spin-echo spectrum with a simple 180° pulse. To investigate the pulse performance in the presence of a resonance offset, the 1H spin-echo spectrum was obtained with a simple 180° refocusing pulse with the transmitter frequency offset by 4.8 kHz from the center of the spectrum (corresponding to a normalized offset parameter DB=Bnom ¼ 0:25) and the 180° pulse 1 length set precisely to its nominal value. The spectrum was phased and the same phase correction then applied to the other experiments performed with this offset with the sequences LF and ASBO-9(B1). The results are shown in Fig. 6c. Perhaps fortuitously, the spectrum recorded with the LF refocusing pulse has the correct phase now but the spectrum acquired with ASBO-9(B1) still has the highest intensity signal at this resonance offset. Experiments were also performed with an offset from resonance and with the constituent elements of the refocusing pulses simultaneously misset. The results are shown in Fig. 6d. Again, the use of LF as a refocusing pulse results in distortion in the phase of the spectrum while the spectrum obtained with ASBO-9(B1) does not have this problem and yields the greatest signal amplitude. 3.3. Comparisons with NMR quantum computing results In recent years, broadband composite pulses have become of interest in the field of NMR quantum computing [27], where they can be used in NMR analogues of quantum logic gates. Working

B1 / B1nom

2.0

in this area, Alway and Jones have proposed a composite 180° pulse [28], made up of nine 180° pulses, that can be written 180°104.5°180°284.5°180°255.5°180°75.5°180°104.5°360°313.4°180°104.5° 180°0° (or its time-reversed form), which we here designate AJ. This pulse is of the phase-distortionless, dual-compensated type and so merits comparison with the pulse TPG from 1985 and with the ASBO-9 family of composite 180° pulses proposed here, both of which have the same overall duration. In a manner analogous to the results in Fig. 4, the simulation in Fig. 7a shows the contour plot of the Iy magnetization in a spinecho experiment with the AJ sequence as the refocusing pulse. As before, contours are drawn at 99%, 95% and 50% of the maximum and plotted as a function of B1 =Bnom and DB=Bnom . As an asymmetric 1 1 sequence, the composite pulse AJ will also produce unwanted outof-phase Ix magnetization (which we do not show here) unlike the antisymmetric sequences ASBO-9 and TPG, which produce no such component. It can be noted that the area enclosed by the 99% contour line in Fig. 7a is smaller than the corresponding areas for the ASBO-9 pulses shown in Fig. 4d and e and for the TPG sequence shown in Fig. 7b. It is interesting to discuss briefly why the AJ sequence performs less well than the TPG pulse and the ASBO-9 family proposed in this paper. First, in part, it is because AJ is not an antisymmetric sequence. However, this does not explain why, even judged purely on the in-phase magnetization component in a spin-echo experiment, as in Fig. 7, it still performs less well than other sequences of the same overall duration. Second, therefore, we must compare the theoretical means by which the AJ sequence and the TPG and ASBO sequences were derived. The AJ sequence was derived using a power series expansion of the error propagator UV(T) (see Eq. (6)). The first term in the expansion of the exponential operator will be linear in the error, the second quadratic, and so on. In contrast, average Hamiltonian theory, as used here to design the ASBO families of pulses, uses a power series expansion of the exact average Hamiltonian V:

U V ðTÞ ¼ expfiVTg

ð29Þ

 ¼ V ð0Þ þ V ð1Þ þ    V

ð30Þ

The first term in the expansion, V(0), is linear in the error but it appears in the error propagator as exp{iV(0)T} and so is already describing nonlinear dynamics. Thus, average Hamiltonian theory converges much faster than direct expansion of the error propagator. This crucial advantage of average Hamiltonian theory has been exploited since its introduction into NMR practice by Waugh and 2.0

(a)

1.5

1.5

1.0

1.0

0.5

0.5

0.0

(b)

0.0 1.0

0.5

0.0

B1 / B1nom

0.5

1.0

1.0

0.5

0.0

0.5

1.0

B1 / B1nom

Fig. 7. Simulations of the in-phase hIyi component yielded by (a) the Alway and Jones pulse 180°104.5°180°284.5°180°255.5°180°75.5°180°104.5°360°313.4°180°104.5°180°0° and (b) the TPG sequence (see Table 1) in a spin-echo experiment as a function of B1 =Bnom and DB=Bnom . The results are presented as a two-dimensional contour plot, with contours 1 1 drawn at 99%, 95% and 50% of the maximum. The simulations assume an initial state rinitial = Iy and a refocusing pulse that has been subjected to the four steps of the Exorcycle phase cycle. The Alway and Jones composite pulse is not antisymmetric and will also yield an unwanted out-of-phase hIxi component (not shown).

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coworkers in the late 1960s [29]. Average Hamiltonian theory now provides much of the theoretical background of modern NMR spectroscopy, especially in solids, and has been used since 1983 to design many composite pulses [8,13,15,17,18,23], including the BB1 family of pulses [18] that has been widely adopted and adapted in the field of NMR quantum computing. 4. Conclusions In this work we have presented the design and evaluation of new antisymmetric broadband composite 180° pulses that are dual compensated, that is, compensated simultaneously for both resonance offset and inhomogeneity of the radiofrequency field. With appropriate selection of the pure spin-echo signal, using phase cycling or magnetic field gradients, these antisymmetric refocusing pulses will produce no phase error across their entire bandwidths whatsoever. This makes them different from previous so-called phase-distortionless composite pulses, based on symmetric or asymmetric phase shift schemes, which will inevitably introduce phase errors into spin-echo experiments. We have designed our new pulses using a graphical interpretation of average Hamiltonian theory. This has yielded an infinite family of 9-pulse antisymmetric refocusing sequences and an infinite family of 11-pulse sequences, all of which are dual-compensated to zeroth- and first-order in the terminology of average Hamiltonian theory. For applications where resonance offsets are small (e.g., 1H NMR), the composite pulse ASBO-9(B1) provides excellent compensation for B1 inhomogeneity and good compensation for resonance offset. Where resonance offsets are larger (e.g., 13 C, 19F, 31P, 29Si NMR), the composite pulses ASBO-9(7A), ASBO9(X) and ASBO-11(X) can all be considered. Finally, it is likely that the ideas introduced here, such as the graphical interpretation of average Hamiltonian theory and the combination of pulse symmetry and coherence transfer pathway constraints, can be exploited in a more general fashion in areas such as magnetic resonance imaging (MRI) and solid-state 2H (I = 1) NMR. Acknowledgments We are grateful to EPSRC for a studentship (S.O.) and to Dr. David Adam for assistance with the 500 MHz NMR spectrometer. References [1] M.H. Levitt, Composite pulses, Prog. NMR Spectrosc. 18 (1986) 61–122.

[2] M.H. Levitt, Composite pulses, in: D.M. Grant, R.K. Harris (Eds.), Encyclopedia of Nuclear Magnetic Resonance, Wiley, Chichester, 1996, pp. 2694–2711. [3] M.H. Levitt, R. Freeman, NMR population inversion using a composite pulse, J. Magn. Reson. 33 (1979) 473–476. [4] R. Freeman, S.P. Kempsell, M.H. Levitt, Radiofrequency pulse sequences which compensate their own imperfections, J. Magn. Reson. 38 (1980) 453–479. [5] M.H. Levitt, Symmetrical composite pulse sequences for NMR population inversion. I. Compensation of resonance offset, J. Magn. Reson. 50 (1982) 95– 110. [6] M.H. Levitt, Symmetrical composite pulse sequences for NMR population inversion. II. Compensation of radiofrequency field inhomogeneity, J. Magn. Reson. 48 (1982) 234–264. [7] J.S. Waugh, Systematic procedure for constructing broadband decoupling sequences, J. Magn. Reson. 49 (1982) 517–521. [8] R. Tycko, Broadband population inversion, Phys. Rev. Lett. 51 (1983) 775– 777. [9] M.H. Levitt, R.R. Ernst, Composite pulses constructed by a recursive expansion procedure, J. Magn. Reson. 55 (1983) 247–254. [10] A.J. Shaka, R. Freeman, Composite pulses with dual compensation, J. Magn. Reson. 55 (1983) 487–493. [11] A.J. Shaka, R. Freeman, Spatially selective radiofrequency pulses, J. Magn. Reson. 59 (1984) 169–176. [12] R. Tycko, A. Pines, Iterative schemes for broadband and narrowband population inversion in NMR, Chem. Phys. Lett. 111 (1984) 462–467. [13] R. Tycko, H.M. Cho, E. Schneider, A. Pines, Composite pulses without phase distortion, J. Magn. Reson. 61 (1985) 90–101. [14] R. Tycko, A. Pines, J. Guckenheimer, Fixed point theory of iterative excitation schemes in NMR, J. Chem. Phys. 83 (1985) 2775–2802. [15] A.J. Shaka, A. Pines, Symmetric phase-alternating composite pulses, J. Magn. Reson. 71 (1987) 495–503. [16] R. Tycko, Iterative methods in the design of pulse sequences for NMR excitation, Adv. Magn. Reson. 15 (1990) 1–49. [17] S. Wimperis, Iterative schemes for phase-distortionless composite 180° pulses, J. Magn. Reson. 93 (1991) 199–206. [18] S. Wimperis, Broadband, narrowband and passband composite pulses for use in advanced NMR experiments, J. Magn. Reson. A 109 (1994) 221–231. [19] S. Odedra, S. Wimperis, Use of composite refocusing pulses to form spin echoes, J. Magn. Reson. 214 (2012) 68–75. [20] D.P. Burum, M. Linder, R.R. Ernst, Low-power multipulse line narrowing in solid-state NMR, J. Magn. Reson. 44 (1981) 173–188. [21] H.M. Cho, R. Tycko, A. Pines, J. Guckenheimer, Iterative maps for bistable excitation of two-level systems, Phys. Rev. Lett. 56 (1986) 1905–1908. [22] H. Cho, J. Baum, A. Pines, Iterative maps with multiple fixed points for excitation of two-level systems, J. Chem. Phys. 86 (1987) 3089–3106. [23] S. Wimperis, Composite pulses with rectangular excitation and inversion profiles, J. Magn. Reson. 83 (1989) 509–524. [24] S. Odedra, S. Wimperis, Improved background suppression in 1H MAS NMR using composite pulses, J. Magn. Reson. 221 (2012) 41–50. [25] G. Bodenhausen, R. Freeman, D.L. Turner, Suppression of artefacts in twodimensional, J. Spectrosc., J. Magn. Reson. 27 (1977) 511–514. [26] G. Bodenhausen, H. Kogler, R.R. Ernst, Selection of coherence-transfer pathways in NMR pulse experiments, J. Magn. Reson. 58 (1984) 370–388. [27] J.A. Jones, Quantum computing with NMR, Prog. NMR Spectrosc. 59 (2011) 91– 120. [28] W.G. Alway, J.A. Jones, Arbitrary precision composite pulses for NMR quantum computing, J. Magn. Reson. 189 (2007) 114–120. [29] J.S. Waugh, Average Hamiltonian theory, in: D.M. Grant, R.K. Harris (Eds.), Encyclopedia of Nuclear Magnetic Resonance, Wiley, Chichester, 1996, pp. 849–854.