JOURNAL OF MAGNETIC RESONANCE 63, 133-141 (1985)

Analytical Theory of Composite Pulses

C. COUNSELL, M. H. LEVITT, AND R. R. ERNST

Laboratorium ftir Physikalische Chemie, Eidgeniissische Technische Hochschule CH-8092 Zurich, Switzerland

Received December 11, 1984

An exact expression for the overall rotation generated by two or three consecutive noncommuting rotations is used for the analysis of several known composite pulse sequences and recursive expansion procedures. Analytical expressions for the dependence of spin inversion on radiofrequency field Strength are given. 0 1985 Academic PIB.S, I~IC.

Despite the considerable success of composite pulse sequences in achieving accurate spin manipulations insensitive to radiofrequency field inhomogeneity or resonance offset effects (l-20), the theory of compos.ite pulses is still incomplete. A variety of theoretical approaches has been used for analyzing composite pulses, but with the exception of numerical simulation, ,none of these have been exact. Geometrical arguments (1-5) have the advantage of simplicity and of providing physical insight, while perturbation theory sacrifices physical directness for generality (6, 13). Another concept which has proved particularly fruitful is recursive expansion (7-10, 16, 19). This allows pulse sequences to be assembled in a hierarchical structure in which the favorable properties of each small building block are not lost as a long sequence with enhanced overall properties is built up.

This diversity of theoretical descriptions is all the more remarkable because the model system which is usually considered is extremely simple. If relaxation and spin-spin or quadrupole interactions may be ignored over the duration of the rf pulse, the spin density operator experiences a pure rotation in a three-dimensional space, and a simple system of two energy levels is sufficiently complex to model the essential features. The equation of motion may be solved exactly and there should be no need to apply perturbation methods. The problem of analyzing composite pulses reduces to the superficially simple task of discovering the overall rotation which is produced by a sequence of two or more component rotations. This is an old mathematical problem whose solution was given by Hamilton in the last century using the quaternion formalism (20). It seems not to have been used much, and only recently have Blilmich and Spiess (21) pointed out the relevance of such exact mathematical solutions to composite pulses. In this paper we work with their equations in an alternative form which avoids the use of matrix multiplication: If a rotation through an angle B1 about the axis nl is followed by a rotation through an angle & about an axis n2, then the overall rotation is through an angle p12 about an axis n12 given by the equations (22)

cl2 = c1c2 - sls2nl - n2 [la1 133 OQ22-2364185 $3.00

Copyright Q 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.

134 COUNSELL, LEVIlT, AND ERNST

s12n12 = slc2nl + cls2n2 - s1:s2nl X n2. [lb1 where ci = cos(@i/%), si = sin&/z), ~12 = cos(fl12/2) and s12 = sin(@r2/2).

A proof of these equations, using spin-j representations of rotations, is given for completeness in the Appendix.

The usefulness of Eqs. [l] will be made clear .by a short example. Consider a rotation by 7r/2 about the y axis followed by a rotation of u/2 about the x axis. In this case cl = sl = c2 = s2 = 22J2, nl = j, and n2 = i. Equations [l] yield immediately

Cl2 = l/2 and

s12n12 = f(i + j + k). PI It follows that /3r2 = 2*/3 and n12 = 3 -12(i + j + k). Thus the overall rotation is by 2n/3 about the (1, 1, 1) axis. This result is hard to derive by other means. It may be verified by applying the composite pulse (?r/2),,(~/2)~ to initial conditions Z,, ZY, and Z, in turn. It is indeed found that this accomplishes a cyclic permutation of subscripts x - y - z - x, etc., which can only be fulfilled by the above rotation.

It is straightforward to extend Eqs. [l] to any number of noncommuting rotations. For example, the three rotations p1 about q, followed by 82 about n2, followed by & about n3, are equivalent to a rotation by /?I123 about the axis 11123, where

Cl23 = ClC2C3 - (sIs2c3nI * n2 + slc2s3nl * n3 + c1s2s3n2 * nj) + sIs2s3(nl X n2) * n3 [3a]

and

s123n123 = slc2c3nl + cls2c3n2 + cIc2s3n3 - (sls2c3nl X n2 + sIc2s3nl X n3

+ w2m X n3) - sls2s3{(n2-n3h - (n3-nh + (nl-n2)n3>. [3bl

The symmetry of Eq. [3a] may be emphasized by noting that (nl X 11~). n3 = nl * (n2 X n3).

APPLICATION TO COMPOSITE PULSES

The rotations which are actually applied to a spin system by a pulse sequence are dependent on the rf field strength and offset from resonance of the irradiation frequency. We denote pulses ($), where @ is the nominal rotational angle, @ = wY~~, WY being the nominal rotation frequency about the on-resonance rf field, 7,, the pulse duration, and 4 the irradiation phase. If the true rf field strength at a given point in the sample is ol, the rotation applied is given by

n = i cos I$ cos 8 + j sin 4 cos 0 + k sin 0 [41 with ti = tan-(Q/o,) and Q is the offset from resonance.

For the rest of this paper we ignore resonance offset effects and consider the effect of on-resonance pulse sequences in an inhomogeneous rf field. The pulse (@), produces a rotation given by

THEORY OF COMPOSITE PIJLSES 135

and n = i cos 4 + j sin C#J ]51

where 6wl = w1 - WY is the deviation of the rotation frequency about the rf field from its nominal value ~0:.

Composite H Pulses

A number of composite pulses have been suggested for population inver- sion insensitive to rf inhomogeneity effects (Z-29). Many of these suggestions are consistent with the formulation ($)O(/3r),($)0. Examples are the composite Pulses (*/2)0(74,,2(d90 (I), (d&@rh,3(d2)0 (4), (dO(d2r/3(d (13, 16 19)~ and (1~)~(7r),,~(7r)~ (Z6). It is remarkable that all of these composite pulses have been treated in different ways. The first was initially derived pictorially (I, 2), later treated by an approximate rotation operator approach (3) and also generated as the first member of a recursive expansion series (10). The second was derived geometrically using antitangential trajectories (4). The third was produced using average Hamiltonian theory (13) and simultaneously presented as an example of a recursive expansion procedure (16). The last was only analyzed by numerical simulation (16). We present below a unified analysis of all of these pulse sequences and give exact expressions for their spin inversion performance at arbitrary rf field strengths.

Equations [ 31 and [ 51 clearly allow one to write down the overall rotations produced by the above pulse sequences in an inhomogeneous rf field. However, some labor may be saved by using symmetry. Let the overall rotation be described by a propagator I!.?

U = exp( ij3J * n,) = exp( $31. n)exp( ifi? . n)exp($I . n) [61 where /$ is the overall rotation angle and n, the elTective rotation axis. It follows that

U-* = exp(-$,I.n,) = exp(-i@I.n)exp(--i@l.d)exp(--$1-n)

and since n and n lie in the xy plane in the absence of off-resonance effects,

exp(-i?rZ,)U- exp(irZ,) = exp(i@I * n)exp(i[lI * n)exp(ipI . n) = U. Hence

[71

PI

-exp( - i?rZJI . %exp(iaZ,) = I - n, . [91

Therefore the effective rotation axis n, also lies in the xy plane. By extension, the result ?f any time-symmetrical sequence of rotations about axes in the xy plane is also a rotation about an axis in the xy plane. This is a convenient property for

The density operator may be rotated by sandwiching it between two operators: u(+) = exp(-iSI.n)~-)exp(igI.n). As in previous work (IO. II, IdI), we consider it more convenient to discuss the operators on the right-hand side of the density operator, since these follow chronological order from left to right and the sign of the exponent is the same as the sign of the sense of rotation.

136 COUNSELL, LEVITT, AND ERNST

analyzing a composite ?r pulse, since only the effective rotation angle & becomes relevant in discussing the inversion of z magnetization: An ideal composite r pulse has & close to 7r over a wide range of rf fields. In general a symmetrical composite pulse produces an inversion of z magnetization given by

(&(+>) = cos 8,. 1101 From Eq. [3a], for a general symmetrical composite pulse (/3-$,(~)&30)o,

Pe P P cos - = cos p cos - - cos f$ sin /I sin - . 2 2 2

Special cases of this equation may now be used to analyze the four composite pulses mentioned above.

(a) The composite pulse (1r/2)~(7r),,&r/2),, (l-3). For this pulse we have p = ?r/ 2, p = ?r, and 4 = 7r/2. Using Eq. [ 111, we find cos(&/2) = cos2(?rw1/2w~) = sin2(7&r/20y). If 6w, $ WY,

2

p.Dr-23 . ( >

iI21

The dependence on 6~: signifies the first-order compensation of rf inhomogeneity effects already remarked (3, 10). For the first time it is possible to give an analytical expression for the degree of population inversion produced by this pulse sequence:

(IA+>) = co42 cos-'(co?(~))~ .

(b) The composite pulse (~/2)d27r)~,,&r/2)0 (4). For this pulse $ = 7r/2, /3* = 27r, and 4 = 2?r/3. Straightforward algebra yields cos(p, /2) = cos3(,,, /2w$ = sin3(x8w, / 204 and if 60, < WY,

This implies second-order compensation of rf inhomogeneity (4). The analytical expression for the dependence of population inversion on rf field is

(1r(+>) = cos{2 cos-l(cos3(~))}. [I51

(c) The composite pulse (?r)O(?r)2&r)0 (13). This sequence, derived originally by average Hamiltonian theory (13) is easily shown analytically to be exactly equivalent to the previous sequence with respect to compensation of rf inhomogeneity.

(d) The composite pulse (lr)O(~),&)O (16). Choosing $ = K, p = K, and 4 = 7~1 2 leads to the equation

cos 2 = cos(~)cos(~)

giving an analytical expression for the population inversion:

(L(+,) = co42 cos-~(cos(~;)cos(~))} [I71

THEORY OF COMPOSITE PULSES 137

Thus population inversion is ideal if wI = WY but also if wI = $o: or $07. This property has been discussed by Shaka et al. (16) and makes the composite pulse (?r)o(~)&?r)o useful as a sequence with coarse compensatory properties over a wide range of rf field strengths.

Detailed examination of Eq. [ 1 l] shows that the composite pulses discussed above are only examples of an infinite number of possible solutions which enjoy first- or second-order compensation of rf inhomogeneity effects. For example, a new solution providing second-order compensation is the composite pulse (2a)o(7r)4(27r)o with 4 = 2 tan.-1($)*2. However, this new sequence has little to recommend it as against known composite pulses. It may also be shown that no symmetrical three-pulse sequences exist with higher than second-order compensation.

Recursive Expansion of Composite ?r Pulses

We previously demonstrated a recursive expansion procedure which generates longer and longer pulse sequences with increasingly accurate compensatory properties (10). The method applies primarily to composite r/2 pulses although composite ?r pulses may be subsequently generated from the 7r/2 pulses (IO). Expansion procedures applying exclusively to composite ?F pulses have since appeared (16, 19). If the composite ?r pulses are denoted R$*), where the overall phase of the composite pulse is 4, these expansion procedures may be formulated

R&*+1 = R &m)R$mR6m). 1181 Thus the composite pulse Rb*+) is three times longer than R&) but has more favorable compensatory properties. We note in pas&g that Shaka et al. (16) gave a rather more complicated formulation of the expansion, involving the inverse sequence (IO), but that Eq. [ 181 is entirely consistent with the sequences they give.

Such expansions arise naturally from the theory above. If resonance offset is neglected, and if firn) are symmetrical in time, then the overall rotation under Rbm) is always about an axis in the xy plane by some overall rotation angle &@. It follows from Eq. [ 111 that

p(em+1, Em p *) cos - = cos pern) cos - - cos t#~ sin &@ sin e . 2 2 2 [I91

Hence Rbm+) achieves a perfect spin inversion (flernl = ?r) if

Lw

or if /3kmJ = 7~. The case 4 = 7r/2 was examined by Shaka et al. (16). In this case p(,m+) = 7~ if

firn) = 7r or p, = ; , g . [211

It follows that the expansion a *+ 1) = @ORW@) WI

138 COUNSELL, LEVITT, ANI) ERNST

leads to a coarse adjustment of the compensation, in which the conditions under which Z?&) performs poorly (so that a, = 7r/2 or 3rr/2 rather than ?r) produce ideal behavior for am+). Shaka et al. (16) demonstrated a nine-pulse sequence generated by two expansion stages of the type of Eq. [22]. Equation [ 191 shows that more general coarse adjustments are possible. For example, if 4 = a/6, then those conditions under which R&) behaves such that @) = 2 COS-(~)~~ produce ideal performance for the expanded sequence a+).

Fine adjustment is also possible by setting 4 = 2rr/3, and using the expansion

R6 m+) = j@)j$)&)~

From the discussion above, in this case

v31

IB(em+o /$B() cos - =

2 cos3 -+ . [241 I,

Thus small deviations in a, from ?r are corrected to second order. A situation which is close to optimal is to pass a starting sequence such as Rp = (alo through a coarse adjustment stage with 4 = a/2 followed by a fine adjustment with 4 = 2*/3. The excellent performance of a nine-pulse sequence generated in this way is illustrated in Ref. (16).

We remark in passing that the expansion procedure of Eq. [23] does not appear to be very much superior to the one previously described (IO). In that case, a twofold expansion of the pulse sequence produces first-order compensation, a fourfold expansion produces third-order compensation, and an eightfold expansion produces seventh-order compensation. In the present case, a threefold expansion yields second-order compensation and a ninefold expansion leads to eighth-order compensation. Very recently, a new expansion procedure has appeared (I 9):

j&m+l) = R6mR6R~~~3Rl;)R~~~3. [251

This needs a fivefold expansion even to produce first-order compensation; however, in contrast to the recipes above, it is also applicable to resonance offset effects.

Composite r/2 Pulses and Composite Pulses of Arbitrary Flip Angle

Once a sequence is generated which is symmetrical in time and which gives a x rotation compensated for rf inhomogeneity, a composite 7r/2 pulse may be derived simply by taking the first half. This is easily seen pictorially (4) or by the following argument: Divide the composite ?r pulse into two halves, A and B, which if the composite pulse is symmetrical, are related to each other by reversal of the order of the component pulses in time. Since A and B consist only of rotations about axes in the xy plane, we may write (see Eqs. [6]-[9])

B = exp(-irZ,)K exp(irZ,). 1261 Furthermore,

B-A-Z&3 = -1 z 1271 since AB is the composite r pulse R&). Hence,

A-ZJ = -BZ,B- = -exp(-i?rZ,)A-ZJ exp(i*Z,). WI

THEORY OF COMPOSITE PULSES 139

But A-ZJ is the result of operating on Z, with A, and Eq. [28] demands that KZJ lie in the xy plane. It follows that A transforms Z, into the xy plane, and hence is a composite ?r/2 pulse. For example, it is possible to derive the composite 7r/2 pulse

(?r)O(?r)=/2(~)O(?r)z*/3(1r/2)7~,6 1291

from the composite 7~ pulse A of Ref. (Z6), Fig. 5. Such a composite ?r/2 pulse could be used to build up composite pulses of arbitrary flip angle as in Ref. (10).

CONCLUSION

The principal merit of the approach demonstrated. in this paper is that it allows the result of several consecutive rotations to be written down analytically and exactly. This allows unified treatment of several previously known pulse sequences which before had had quite different theoretical descriptions. We do not wish to hide the fact that exact analysis in this way becomes cumbersome when more than three rotations are introduced or when rotations about tilted effective fields are included. In that case the approximate methods presented in Refs. (Z-19) are indispensible. Nevertheless we feel that exact analysis is a useful addition to the array of tools used in studying composite pulse sequences, and further applications can be conceived.

APPENDIX

In this appendix we give a proof of eqs. [la] and [lb]. For convenience we work with the spin-4 representation of the rotation group. This in no way implies that Eqs. [I] are restricted to spin-$ systems since they relate physical characteristics of rotations to each other (angles and axes), which are independent of the representation.

We will make use of the Kronecker delta 6, and the permutation symbol expu (23, 24), where A, ~1, u = x, y, z. The 6 symbol has the property a,, = 1 if X = p but is zero otherwise. The symbol eXrv is zero if two subscripts are equal and is +I otherwise; it is invariant to cyclic permutation of the three subscripts but changes sign if two are interchanged, and by convention fxYr = 1. We shall also use the notation that if a subscript appears twice on one xide of the equation then this implies that a summation is to be made over that subscript (24). In this way the dot and cross products of two vectors can be written as

and

(A X B) - C = A~B,,C&.

The familiar commutation relationships between angular momentum operators Z,, I,,, and Z, become

[IA, Z,] = Z,Z, - Z,I, = &Iv. 1311

For the spin Z = f representation, the anticommutation relationships may be written

1321

140 COUNSELL, LEVITT, AND ERNST

Equations [3 I] and [32] can be combined to give an expression for the product of two operators in this representation:

]331

We will need an expression for the more general product (I * n,)(I * n2) where

I = 1,i + Z,,j + I,k ]341

ni = ni$ + ni,J + nizk

and

n~+n$+n;z= 1.

This may easily be evaluated:

(1. n,)(I . n2) = 6A~6~gzlAn2k~ZpI,~ = BxrS~,~r~1Xn2X,(~i~~~,~~ + f6,, 1)

= ~~~~~~~~~~~~~~~ + $3,,n,,n2, ll) = $(nl X nz).I + gn, .n,)l [35]

using Eqs. [30] and [33], and the repeated subscript convention. We now use Eq. [35] to prove Eqs. [l]. A general rotation through an angle p

about the unit vector n may be represented by the operator

exp{@I.n}. ]361

The exponential can be expanded giving, in the I = 4 representation,

exp($I . n) = n cos(p/2) + 2iX - n sin(P/2). 1371

This follows because from Eq. [35] (1-n) = ill. To determine the product of two rotations B1 about nl followed by fi2 about n2, we form the equation

exp{i~121.n12) = exp{i&I~nl}exp{i/321~n2}. ]3gl On expansion of both sides,

cl2 + 2isi21 - n12 = clc2 + 2i{slc21. nl + c,s;,I . n2} - 4S&(I * nl)(I + n2) ]391 with cj and si defined above. Substituting Eq. [35] yields

cl2 + 2is121 - n12 = clc2 - (n, * n2)s1s2 + 2i{slc2n, t c1s2n2 - sls2(nl X n2). I [40]

from which the real and imaginary parts can be identified separately to give Eqs. [la] and [lb].

ACKNOWLEDGMENTS

This research was supported by the Swiss National Science Foundation. C.C. acknowledges a Royal Society European Science Exchange Program Fellowship. We also thank B. Bliimich and H. Spiess for providing Ref. (24 prior to publication, and for useful discussions.

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THEORY OF COMPOSITE PULSES 141

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