JOURNAL
OF MAGNETIC
RESONANCE
93, 199-206
( 1991)
Iterative Schemes for Phase-Distortionless
Composite 180” Pulses
STEPHEN WIMPERIS Department of Chemistry, University of Manchester, OKford Road, Manchester Ml3 9PL, United Kingdom Received
December
20, 1990
Early enthusiasm for broadband composite 180” pulses was tempered when it became known that such sequences do not always show an advantage over simple 180” pulses when used for refocusing of transverse magnetization. For example, the first (and still most widely used) composite 180” pulse, 90& 18OP~0~90& (I), achieves broadband inversion of z magnetization with respect to the value of the radiofrequency field strength of the pulses. When used for refocusing of transverse magnetization, however, this composite pulse produces an echo signal with broadband amplitude characteristics but with a phase that is extremely sensitive to the precise value of the pulse field strength (2). Clearly this is a major disadvantage in those NMR experiments (e.g., INEPT) which require a specific relationship between the phase of the echo signal and the phase of the next radiofrequency pulse. Even worse, however, if the deviation of the actual pulse field strength from its nominal value is due to spatial inhomogeneity of the radiofrequency field rather than a simple miscalibration, then the superposition of many echo signals with different phases will result in partial or even complete cancellation of the total signal in any spin-echo experiment. The problem with a composite 180” pulse such as 90& 180 i’S0090&0is therefore that, although its overall flip angle is broadband with respect to the value of the radiofrequency field strength, its overall rotation axis is strongly dependent upon this parameter. In this respect the composite pulse 90,“,. 180 ~s0~90&,0is the opposite of a simple 180” pulse which has a flip angle that is linearly dependent upon the field strength but a rotation axis that is constant. As a result of these considerations, there has been much interest in recent years in the design of phase-distortionless composite pulses (3-8). 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. This Communication presents novel phase-distortionless composite 180” pulses which are broadband with respect to the value of the pulse field strength and can therefore be used for accurate refocusing of transverse magnetization in the presence of severe spatial inhomogeneity of the radiofrequency field. Both the five-pulse sequences presented here owe their excellent performance to the use of symmetry properties in their design and can be employed, if the need for even superior performance arises, as the starting points for jteratjve expansion schemes (5, 7-15) that generate longer, more refined composite pulses. Some steps used in the design of these new composite 180” pulses are similar to those 199
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used previously in the design of rectangular-profile composite pulses (14) and broadband composite 90” pulses ( 15) and so will only be outlined here. Resonance offset effects, as in previous studies, will be deliberately neglected throughout the following discussion. We consider a composite pulse made up of a sequence of n nominal 180” pulses: 1SO;, 1SO;, . * * 180&
ill
The advantages of constraining any new composite 180” pulse to have this form will emerge in the course of this discussion. The propagator U(p) for this sequence can be written U(p) = rI exp { -i/U,, >,
j=n
[21
where Z$, = Z&OS 4, + Z,sin ~j. Note the left to right antichronological order of the exponential operators. The actual flip angle of the individual pulses ,L3= w,7,800 can bewritten/3=~+6,where6=(wl-WY”‘) r, 800is the deviation of p from its nominal value (7r or 180” ) . The propagator U( /3) can now easily be separated into two propagators U( 7r) and fi( 6)
FIG. 1. Simulated performance at exact resonance of a nominal 1SO& pulse (dotted line) and the previously proposed composite 180” pulses 90& 180&,.90&( dashed line) and 1SO& 180&~180~200 (solid line) as a function of the normalized nutation frequency w , /w ym. The figure shows the coefficients of I, (top) and I, (bottom) resulting from application of the sequences to the initial conditions a(O) = Zz and u(O) = I,, respectively. Both composite 180” pulses show broadband characteristics for the inversion of z magnetization, but only the phase-distortionless sequence 180&e 1SO&,. 180pzo0shows broadband characteristics for the transformation I, + - IY (the dashed line has coalesced with the dotted line in the bottom plot).
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1.0 \ \ Iz
\\ ‘1, \
-1.o IL
\
/ \\
-J
I
I
//
/
/
/
,1 1
/ /’
1.0
IY
-1.0
1.0
0
;.O
FIG. 2. Simulated I, + -I, (top) and I), + -I, (bottom) profiles of a nominal I80& pulse (dotted line) and the new composite 180” pulses F, of Eq. [IO] (dashed line) and FZ of Eq. [ 121 (solid line) as a function of the normalized nutation frequency wI /w;“” at exact resonance. Note that the extremely close similarity between the profiles for the two orthogonal transformations is an indication that both composite 180” pulses have rotation axes that are essentially coincident with the rotating-frame x axis for all values of w, /a;““.
U(P) = U(dfiW,
[31
U(a) = II exp{ -kr&} ,=n
141
O(S) = Ii exp(-iU$l,) ,=n
[51
where
and
with 4; = -(-I)‘&
/-I - c (-l)k2&.
X=l
[61
If II is odd then U( 7r) is the propagator for a perfect composite 180” pulse whatever the values of the phases { 4,}. The propagator o(S), which effectively acts before U(x) in time, describes the nonideal performance of the composite pulse. In the language of average Hamiltonian theory, the phases { 4j} that occur in o(S) are the pulse phases { $,} transformed into the toggling frame imposed by U(a).
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1.0
IZ -1 .o
--.
0
1.0
2.0
01 /wnom FIG. 3. Simulated I, + -I, (top) and IY + - IY (bottom) profiles of a nominal 1SO& pulse (dotted line) and the new composite 180” pulses G, of Eq. [ 131 (dashed line) and G2 of Eq. [ 151 (solid line) as a function of the normalized nutation frequency w, /my”’ at exact resonance.
All that is necessary to design a phase-distortionless broadband composite 180” pulse is to choose values for 12and the phases 4, to 4, such that for small values of 6 the propagator I?( 6) x 1. The full composite-pulse propagator U(p) will then approximately equal the ideal propagator U(X) over this range of 6. This aim can be achieved using the average Hamiltonian method of Tycko (3) and expanding the propagator O(F) as a power series in 6: l?(S) = exp{ --is co),co).I - i~‘l’n”’ .I - i~‘2’n’2’ .I - . . . } [71 with the zeroth-order average rotation
the first-order average rotation
and so on. For small 6 the only significant term in the expansion of Eq. [ 7 ] is 6 Co)nco). I and if this can be set equal to zero by a judicious choice of values for n (which must be odd!) and for the phases { 4;;) then the resulting phase-distortionless
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FIG. 4. Experimental inversion profiles for (a) a Eq. [lo]. and (c) the composite 180” pulse G, of further experimental details. These profiles do not and 3 because a change in flip angle does not mimic field when. as in this case. there is a genuine spatial
simple 180” pulse, (b) the composite 180” pulse F, of Eq. [ 131 as a function of pulse flip angle. See text for correspond exactly to the simulated profiles of Figs. 2 exactly a spatial inhomogeneity of the radiofrequency inhomogeneity present as well.
composite 180” pulse will be broadband to zeroth-order in the radiofrequency field strength. For n = 3, a solution to the equation 6 (‘)n”) . I = 0 can be found with +‘, = 120”. 4; = O”, 4; = 240”. This analytical solution can readily be verified graphically. An easy mistake to make now would be to assume that the sequence 180$00180&180& is a broadband composite 180” pulse when in fact, quite to the contrary, it is a well-known narrowband composite 180” pulse ( 10, 11). The error arises because the phases { 4;) are the toggling-frame phases and not the actual pulse phases { 4,}. The latter must be calculated from the { $J} using Eq. [ 61 and are given by til = I20”, & = 240”) & = 120”. Therefore the sequence 180Y100180& 18OP,o. is a phase-distortionless composite 180” pulse broadband to zeroth-order. However, this is also an existing composite pulse, being first derived by Tycko using an average Hamiltonian approach in 1983 (3). Figure 1 compares the computer-simulated performance of 18O;;O~180”24001SOP,,. to the performance with respect to the value of the radiofrequency field strength of 90t00 180~so090~00and a simple nominal 180,$ pulse. All composite 180” pulses described in this Communication are written as nominal rotations about the rotating-frame x axis and so, if truly broadband and phase-distortionless, should have broadband profiles for both transformations IZ -P -ZZ and !), + -I,.. The figure shows that only 180pZo01802q100 180”rIoOis successful in this respect. A significant improvement upon the performance of the sequence 180~zo0 180&00 180Pzoacan be achieved by using the well-known property that all odd orders of average Hamiltonian vanish if the exact Hamiltonian (in the toggling frame) is
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time-symmetric ( 16). For the special case discussed here this translates into the result that all odd orders of average rotation 6 (2N”)n(2N+1) * I are zero if the phases { c$;} are symmetric (i.e., if c$: = $L+l-,). This can clearly be seen to be true for the case of the first-order average rotation of Eq. [ 91 where, for example, [I+;, I,;] = - [I,;, I$;-,] if C#JJ = $L+rei. An analytical solution to the equation 6 “)n”) . I = 0 withoddnandsymmetric{$j)canbefoundwithn= 5and4’, =c$;=~cos-‘(0.25) - cos-‘(-0.25), q5; = 4; = 4 cos~‘(O.25) - cos-‘(-0.25), $5 = 4 cos~‘(O.25) (or approximately c$‘,= 46.6”, ~$5= 197.6”, +j = 302.1”, c#J!,= 197.6”, 4; = 46.6”) and, again, an excellent (but previously proposed (5)) narrowband composite 180” pulse is the result if these phases are treated as pulse phases. Using Eq. [ 61 the actual pulse phases can be calculated as 4r = 46.6”, $2 = 255.5”, C$Q= O”, & = 104.5”, c#+ = 3 13.4”. Thus the broadband composite 180” pulse F, = 180&” 180&a 180& 18OPo4.5” 180;,3.4”
[lOI
has6~0~n~0~~I=6~‘~n~‘~~I=O(andinfactal16 (2Nt1)n(2N+‘) . I = 0) and so is broadband and phase-distortionless to first-order in the radiofrequency held strength, The fact that a composite 180” pulse made up only of nominal 180” pulses, such as F, , has symmetric toggling-frame phases { 45 > automatically ensures that the actual pulse phases ( 4j} are antisymmetric: i.e., the composite pulse is of the form lSO,O,180;1. . . 1f$();o.. ~180~,,18OY?,, found in Eq. [lo]. It is known that antisymmetric composite pulses have overall rotation axes that are constrained to lie in the xz plane of the rotating frame whatever the actual flip angles of the individual pulses (5). This is in contrast to symmetric composite pulses, such as 90,0,. 180~so0909”,0 or l8Op200180iao0 18Oy200,which have overall rotation axes that lie in the xy plane ( 17). It has been argued that symmetric composite pulses are well-suited to broadband inversion of z magnetization since inversion will be achieved whatever the direction of the rotation axis in the xy plane provided that the overall flip angle is 180” (17). Therefore it can be argued similarly that antisymmetric composite pulses are better suited to broadband refocusing of transverse magnetization since, with an overall flip angle of 180”) they will achieve the transformation ZY--) - ZYwhatever the direction of the rotation axis in the xz plane (5). The composite 180” pulse FI of Eq. [lo] is therefore predicted to be a refocusing sequence superior to 180~~~~180~~~~18OP~~~ on two counts: it is broadband and phase-distortionless to higher-order in the radiofrequency field strength, and it is a rotation in the xz rather than the xy plane. The latter property means that even when the rotation axis of F, does deviate from the rotating frame x axis the effect on the phase of the echo signal is less severe than with an equivalent symmetric composite pulse. The composite pulse F, is only the first member of a family of phase-distortionless broadband composite 180“ pulses { Fk} which can be generated by iterative expansion. If OF,(S) is the toggling-frame propagator (see Eqs. [ 3 ] - [ 61) for the composite pulse Fr , then the propagator gFk+,(S) for the (k + 1) th member of the family can be written
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Thus this iterative scheme consists of successively repeating in the toggling frame the pattern of phases { 45} that were used to construct F, originally (15). Applying this iterative scheme once, and calculating the actual pulse phases { $j} from Eq. [ 61, one finds the broadband composite 180” pulse: F2 = 1SO~~3.,~18O~~2,,~18O~~6.6~18O;;,~18O~~18O~~o9~18O~~l8O~~55.5~18O~s~~ 180&2,1” 180&” 180;&= 180& 180$4,50 180&4~ 180&,~ 180;~ 1800”~180p~,~1800”~180~~09~18030,3.4”180507.9”180~~66.9”. [ 121 This composite 180” pulse has 6 co,,co,.~ = ,j(l$,(‘).~ = ~(*+,(2).1 = &(3)n(3).~ = 0 and so is broadband and phase-distortionless to third-order in the radiofrequency field strength. The next member of the family, F3, will consist of 125 pulses and be broadband and phase-distortionless to fifth-order. Note that the toggling-frame phases { 4; } remain symmetric and the pulse phases { 4, } remain antisymmetric for all members of the family { Fk } . Figure 2 compares the computer-simulated Zz + -Z, and ZY+ -ZY profiles for the composite 180” pulses F, of Eq. [lo] and F2 of Eq. [ 121 to that for a simple nominal 1SO&pulse. The excellent broadband and phase-distortionless properties of the { Fk} family of composite pulses are apparent. A second and closely related family of phase-distortionless broadband composite 180” pulses { Gk > can be designed by a variant of the approach used for the family { Fk } ( 10, 14, Z-5). If values for y2and the phases { 4;) can be found such that the toggling-frame propagator U( 6) of Eq. [ 51 exactly equals unity when 6 = 7r/ 2, then the full composite-pulse propagator U( 0) will exactly equal the ideal propagator U( rr) when p = r/2, s, and 37r/2. It can then be assumed that U(p) N U( 7r) for all values of p between r / 2 and 3~12. This can be achieved with n = 5 and with the symmetric toggling-frame phases cb’, = 45”, 4; = 180”, 4; = 270”, 4h = 180”, c$; = 45”. The actual pulse phases { $j} can be calculated from Eq. [6] and give the broadband composite 180” pulse: 1Wo4.5~
G, = 180&a 18O;,o” 1SO;. 1SOto,.1809,5”.
[I31
The iterative expansion scheme that generates the family { Gk} again consists of successively repeating in the toggling frame the pattern of phases { 4J} that were used to construct G, . Thus the propagator 0 Gk+,(S) for the (k + 1)th member of the family can be written &,+,(s)
=
[irc,(b)145”[~~k(~)11800[iiG‘k(~)1270”[~~k(~)l1800[iick(~)1459
[I41
and if this iterative scheme is applied once it gives the phase-distortionless broadband composite 180” pulse: G? = l8O~~~l8O,O~~~l8O~~~l8O~3~~180~~180~~~~180~~180~~~~180~~~~ 18030,5”1SO& 180;,70”1SO&1SOto,.180& 1SO& 1SO&,0 18o~~~18oo”~18op35”18o~“l8o~~25”18o3o,5”18o~*~18o24o~. [ 151 Once again note that the toggling-frame phases { 4; > remain symmetric and the pulse phases { 4, } remain antisymmetric for all members of the family { Gk > , Figure 3
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compares the computer-simulated Z, + -1, and IY + -IY profiles for the composite 180” pulses G, of Eq. [ 131 and G2 of Eq. [ 151 to that for a simple nominal 180& pulse. It can be seen that members of the { Gk} family have a broader bandwidth than the equivalent { Fk} composite pulses but are less accurate within this bandwidth. The experimental inversion profiles for (a) a simple 180” pulse, (b) the composite 180” pulse F, of Eq. [lo], and (c) the composite 180” pulse G1 of Eq. [ 131 are shown in Fig. 4. A heavily “doped” sample of chloroform in CDC13 ( T1 = 60 ms) was used. Each of the three series of 19 spectra was obtained by incrementing the nominal flip angles in the inversion pulses from 180” to 360” in 10” steps while monitoring the extent of inversion with an appropriately phase-cycled 90” “read” pulse. The predicted broadband behavior of sequences F, and G, is confirmed. These spectra were recorded on a Bruker AC-300E spectrometer equipped with a Varian magnet. The phase shifts were set as multiples of ( 360 / 128 ) ‘; this is easily sufficient accuracy for the composite pulse F1. In conclusion, two families of phase-distortionless composite 180” pulses, { Fk} and { Gk} , which are broadband with respect to the value of the radiofrequency field strength have been designed. Although the higher-order composite 180” pulses F2, F3,... and G2, G3,. . . , etc., are probably too unwieldy for many applications, the simplest sequences, F, of Eq. [ 10 ] and G, of Eq. [ 13 1, are already superior (as a result of their symmetric toggling-frame Hamiltonians) to previously proposed sequences of comparable length. They should therefore prove useful in situations where there is a severely inhomogeneous radiofrequency field, such as is produced by a surface coil. ACKNOWLEDGMENTS
The spectrometer used in this work was purchased with the aid ofgrants from the Science and Engineering Research Council and the University of Manchester. REFERENCES 1. M. H. LEVITT AND R. FREEMAN, J. Magn. Reson. 33,473 ( 1979). 2. R. FREEMAN, S. P. KEMPSELL, AND M. H. LEVIS, J. Magn. Reson. 38,453 (1990). 3. R. TYCKO, Phys. Rev. Lett. 51, 775 (1983). 4. R. TYCKO, H. M. CHO, E. SCHNEIDER, AND A. PINES, J. Magn. Reson. 61,90 ( 1985). 5. R. TYCKO, A. PINES, AND J. GUCKENHEIMER, J. Chem. Phys. 83,2775 ( 1985). 6. A. J. SHAKA AND A. PINES, J. Magn. Reson. 71,495 (1987). 7. H. CHO AND A. PINES, J. Chem. Phys. 86, 6591 (1987). 8. M. H. LEVITT, Prog. NMR Spectrosc. 18, 61 (1986). 9. M. H. LEVITT AND R. R. ERNST, J. Magn. Reson. 55,247 ( 1983). 10. A. J. SHAKA AND R. FREEMAN, J. Magn. Reson. 59, 169 ( 1984). 11. R. TYCKO AND A. PINES, Chem. Phys. Lett. 111,462 (1984). 12. H. M. CHO, R. TYCKO, A. PINES, AND J. GUCKENHEIMER, Phys. Rev. Lett. 56, 1905 ( 1986). 13. H. CHO, J. BAUM, AND A. PINES, J. Chem. Phys. 86,3089 (1987). 14. S. WIMPERIS, J. Magn. Reson. 83, 509 (1989). 15. S. WIMPERIS, J. Magn. Reson. 86, 46 (1990). Resolution NMR in 16. U. HAEBERLEN, “Advances in Magnetic Resonance,” Suppl. 1, “High
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Solids,”