Broadband 180 degree universal rotation pulses for NMR spectroscopy designed by optimal control
Thomas E. Skinner, Naum I. Gershenzon, Manoj Nimbalkar, Wolfgang Bermel, Burkhard Luy, Steffen J. Glaser
aa r X i v : . [ phy s i c s . c h e m - ph ] N ov Broadband 180 ◦ universal rotation pulsesfor NMR spectroscopy designed by optimal control Thomas E. Skinner a, ∗ , Naum I. Gershenzon a , Manoj Nimbalkar b , WolfgangBermel c , Burkhard Luy d,e , Steffen J. Glaser b a Physics Department, Wright State University, Dayton, OH 45435, USA b Department of Chemistry, Technische Universit¨at M¨unchen, Lichtenbergstr. 4, 85747Garching, Germany c Bruker BioSpin GmbH, Silberstreifen 4, 76287 Rheinstetten, Germany d Institut f¨ur Organische Chemie, Karlsruher Institut f¨ur Technologie, Fritz-Haber-Weg 6,76131 Karlsruhe, Germany e Institut f¨ur Biologische Grenzfl¨achen 2, Karlsruher Institut f¨ur Technologie, 76344Karlsruhe, Germany
Abstract
Broadband inversion pulses that rotate all magnetization components 180 ◦ about a given fixed axis are necessary for refocusing and mixing in high-resolution NMR spectroscopy. The relative merits of various methodologiesfor generating pulses suitable for broadband refocusing are considered. Thede novo design of 180 ◦ universal rotation pulses (180 ◦ UR ) using optimal con-trol can provide improved performance compared to schemes which constructrefocusing pulses as composites of existing pulses. The advantages of broad-band universal rotation by optimized pulses (BURBOP) are most evident forpulse design that includes tolerance to RF inhomogeneity or miscalibration.We present new modifications of the optimal control algorithm that incor-porate symmetry principles and relax conservative limits on peak RF pulseamplitude for short time periods that pose no threat to the probe. We applythem to generate a set of 180 ◦ BURBOP pulses suitable for widespread use in C spectroscopy on the majority of available probes.
Keywords: refocusing pulses; universal rotation pulses; UR pulses;BURBOP; optimal control theory ∗ Corresponding author.
Email addresses: [email protected] (Thomas E. Skinner), [email protected] (Steffen J. Glaser)
Preprint submitted to Elsevier November 30, 2011
ACS:
1. Introduction
Many NMR applications require refocusing of transverse magnetization,which is easily accomplished on resonance by any good inversion pulse sand-wiched between delays, ie, the standard ∆–180 ◦ –∆ block. For broadbandapplications, a universal rotation (UR) pulse that rotates any orientation ofthe initial magnetization 180 ◦ about a given fixed axis is required to refocusall transverse magnetization components. A simple hard pulse functions asa UR pulse only over a limited range of resonance offsets that can not beincreased significantly due to pulse power constraints.Although a great deal of effort has been devoted to increasing the band-width of inversion pulses, most broadband inversion pulses [1, 2, 3, 4, 5, 6,7, 8, 9, 10, 11, 12, 13, 14, 15, 16] execute only a point-to-point (PP) rota-tion for one specific initial state, magnetization M z → − M z , and are notUR pulses. However, two PP inversion pulses with suitable interpulse delayscan be used to construct a refocusing sequence [17, 18], which is effectively a360 ◦ UR pulse. Alternatively, a 180 ◦ UR refocusing pulse can be constructed fromthree adiabatic inversion pulses [19] with either pulse length or bandwidthof the adiabatic frequency sweep in the ratio 1:2:1. More generally, we haveshown that one can construct a UR pulse of any flip angle from a PP pulseof half the flip angle preceded by its time- and phase-reversed waveform [20].Thus, a 180 ◦ UR pulse can be constructed from two 90 ◦ P P pulses.The reliance on composites of PP pulses to construct UR pulses high-lights the perceived difficulty of creating stand-alone UR pulses. The denovo design of UR pulses for NMR spectroscopy has received compara-tively little attention [9, 21], so it is an open question whether the com-posite constructions using PP pulses achieve the best possible performance.Yet, the demonstrated capabilities of optimal control for designing PP pulses[22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] are equally applicable tothe design of UR pulses [36, 37, 38]. The required modifications to the basicoptimal control algorithm are fairly straightforward [36, 39, 40] and maintainthe same flexibility for incorporating tolerance to variations in experimentallyimportant parameters, such as RF homogeneity or relaxation.In this work, we design broadband refocusing pulses by optimizing thepropagator for the required UR transformation. The resulting 180 ◦ BURBOP pulses (broadband universal rotation by optimized pulses) are compared to2xisting composite refocusing pulse schemes to characterize the conditionsunder which one design method might be preferrable to another. In addition,we introduce new optimal control strategies tailored to take advantage ofspecific opportunities available in the design of UR pulses. The culminationof these efforts is a set of low-power, high-performance broadband refocusingpulses that satisfy the power constraints of widely available probeheads andcomplex multipulse sequences.
2. Optimal control algorithm for 180 ◦ UR pulses A general procedure for creating a desired unitary propagator in an arbi-trary (closed) quantum system is given in [36, 39, 40, 41]. Time evolutionproceeds according to a matrix exponential of the system Hamiltonian. Fortwo-level systems, as in many NMR applications involving a single nonin-teracting spin-1/2 species, this evolution is well-known to be equivalent toa rotation of the 3D vector representing the state of the system about theeffective applied field [42]. The relatively abstract general procedure forpropagator optimization can be made considerably more transparent in thiscase.
The optimal control methodology for generating PP transformations intwo-level systems has been described in detail previously [22, 23, 24, 25, 26,27, 28, 29, 30, 31, 32, 33, 34, 35], with progressive modification to enhancepulse performance by incorporating various experimental constraints. In eachiteration of the algorithm, one starts with a given initial magnetization M ,applies the RF derived for the current iteration, and compares the resultingfinal state M f with a desired target state F . This comparison, quantifiedin terms of a cost function, allows one to efficiently calculate a gradient forimproving pulse performance in the next iteration. A simple but effectivecost function Φ, which we employ in the present work, is the projection ofthe final state onto the target state, the standard inner product h F | M f i ,given in this case by the dot product F · M f . Desired RF limits can beenforced by clipping without degrading the performance of the optimization[28].The algorithm for generating UR pulses in the single-spin case is a straight-forward modification of the PP algorithm. A 180 ◦ UR pulse applied, for exam-ple, along the y-axis to magnetization M effects the transformation ( M x , M y , M z ) → − M x , M y , − M z ). This is simply three separate PP transformations of theinitial states M = (1 , , M = (0 , , M = (0 , ,
1) to their respec-tive target states F = ( − , , F = (0 , , F = (0 , , − M kf ( k = 1 , ,
3) at the end of an RFpulse to the target states isΦ = F · M f + F · M f + F · M f . (1)The algorithm proceeds in the standard fashion using this cost function. Wewill refer to this as algorithm A.This simple intuitive modification to the cost is exactly equivalent to ananalogous procedure given in [36] for optimizing the unitary propagator,which can be seen as follows. The rotation operator R F in 3D correspondingto the target propagator that generates a 180 ◦ rotation about the y -axis isgiven by R F = − − = ... ... ... F F F ... ... ... , (2)ie, the i th column is the corresponding PP target F i .The actual rotation operator at the end of a pulse of length T p is R ( T p ) = R ( T p ) = R ( T p ) ... ... ... M M M ... ... ... = ... ... ... M f M f M f ... ... ... , (3)with the rotation transforming each column to its associated final state forthe individual PP transformations.The cost is again given by the projection of the final state onto the targetstate, with the inner productΦ R = h R F | R ( T p ) i = Tr [ R T F R ( T p )] , (4)4here superscript T denotes the transpose, and the operator Tr returns thetrace (sum of diagonal elements) of its argument. We then have R T F R ( T p ) = · · · F · · ·· · · F · · ·· · · F · · · ... ... ... M f M f M f ... ... ... . (5)The sum over diagonal elements of this matrix product gives Eq. [ 1 ]. The formalism for constructing UR pulses from PP pulses [20] providesadditional insight for improving 180 ◦ BURBOP performance. The symmetry ofthe construction procedure constrains the resulting rotation axis to the planedefined by the desired axis and the z -axis [43]. Details of the results whichfollow are provided in the Appendix.For a UR rotation about the x -axis, any nonidealities in the original PPpulse (for example, due to resonance offset) shift the resulting rotation axisonly in the xz -plane. Given a phase deviation of magnitude δφ from thedesired target state and rotation error δθ compared to the desired rotationangle, the angle α UR that the rotation axis in the xz -plane makes with the x -axis is small and bounded according to the relation α UR . p ( δφ PP ) + ( δθ PP ) (6)for small deviations measured in radians.This has the effect of reducing any phase errors in the original 90 ◦ P P pulseused for the construction. For example, a 180 ◦ rotation about any axis inthe xz -plane transforms I y to − I y with no phase error. There would be nophase error in the I x → I x transformation, either, but the amplitude of thefinal x -magnetization would decrease depending on the angle α UR , givinga cos(2 α UR ) dependence. Similarly, the I z → − I z transformation wouldhave a cos( α UR ) dependence for the magnitude of the final z-magnetization.Phase errors are thus the result of deviations, δη , from the ideal 180 ◦ rotationangle, which are bounded (see Appendix) according to δη . p ( δφ PP ) + ( δθ PP ) (7)Larger amplitude errors result from both the displacement of the rotation axisfrom the x -axis and deviations from the ideal 180 ◦ rotation angle. Phase de-viations in the 90 ◦ P P pulse are reduced in the resulting 180 ◦ UR pulse according5o the relation δφ UR . ( δφ PP ) + ( δθ PP ) . (8)The reduction is quite significant, with δθ PP = 1 ◦ and δφ PP of 10 ◦ , 3 ◦ ,and 1 ◦ , for example, giving bounds for δφ UR of 1.76 ◦ , 0.17 ◦ , and 0.03 ◦ ,respectively.For applications requiring high phase fidelity which can afford modestloss of signal intensity, we therefore incorporate the symmetry principle ofthe construction procedure into the optimal control algorithm. For RF pulsecomponents u x and u y digitized in N time steps, the first half of the pulse isdetermined using the basic algorithm A. The second half of the pulse is thenconstructed using the time-and phase-reversed components of the first half.Phase zero for u x leaves it unaffected, while u y is inverted to give u i + N/ x = u N/ − ix u i + N/ y = − u N/ − iy (9)for i = 1 , , , . . . , N/
2. We refer to this algorithm incorporating the symme-try of the construction principle as A S . Peak RF amplitude must remain below probe limits (e.g., available for C spectroscopy), but larger RF amplitudes result in improved broadbandperformance. For sufficiently short time periods, we note that probe RFlimits can be higher than conservative limits that protect the probe fromarcing under any conditions. Enforcing a lower probe limit for an entire pulseduration can sacrifice performance unnecessarily. We therefore introduce atime-dependent RF field limit to allow increased RF amplitude for short timeintervals and achieve improved performance. Empirically, we find that lowRF limits force algorithm A to request higher RF amplitude in the middleof the pulse for improved performance. We therefore allow a higher RF limitfor a short time during the middle of the pulse. We refer to this algorithmas A T , or, if it is also combined with the symmetry principle, as A S,T .
3. BURBOP compared to refocusing with PP pulses
Performance of the new 180 ◦ BURBOP pulses designed using optimal controlare compared with previous methods for generating refocusing pulses, start-ing with those that can be constructed from 90 ◦ P P pulses. We then consider6he refocusing performance of two 180 ◦ P P pulses. In both these comparisons,RF amplitudes are limited to a relatively conservative peak value of 11 kHz(22.7 µ s hard 90 ◦ pulse) for widespread use in C spectroscopy. Finally,we present a set of 180 ◦ BURBOP pulses that utilize the higher RF power lim-its allowed for short time intervals to increase the maximum RF amplitudefrom 11 kHz to 15 kHz (16.7 µ s hard pulse) in the middle of the pulse. Theperformance of these pulses is compared with the composite adiabatic pulsescheme [19] implemented as a smoothed Chirp pulse [44, 45] in standardBruker software . S ◦ P P pulses
A previously published 90 ◦ P P pulse with exceptional performance [30] wasused to construct a 180 ◦ UR pulse according to the procedure in [20]. This 1 msconstant amplitude phase-modulated excitation pulse transforms greater than99% of initial z-magnetization to the x-axis over a resonance offset range of50 kHz for RF amplitude anywhere in the range 10–20 kHz. For all butthe lowest RF amplitudes, the transformation is greater than 99.5%. Phasedeviations of the excited magnetization from the x-axis are less than 2 ◦ –3 ◦ over almost the entire optimization window, with minor distortions in the6 ◦ –9 ◦ range at the lowest RF values.The upper panel of Fig. 1 depicts the performance of the constructed pulsecomposed of two 1 ms 90 ◦ P P pulses. As expected from the discussion in § ◦ P P pulse used for the UR con-struction in Fig. 1a. Thus, algorithm A maximizes signal amplitude, withphase fidelity secondary, while the construction procedure maximizes phasefidelity, with signal amplitude secondary.When the optimization includes a range of RF inhomogeneity/miscalibration,algorithm A can produce shorter pulses than the construction principle for a7omparable amplitude performance, as shown in Fig. 1c, at the cost of poorerphase performance. We find empirically that the reduction in pulse length is50%, 30%, and 15% for RF miscalibration ranges of ± , ± ± S utilizes the construction procedure symmetry inoptimizing the UR propagator, producing the pulse performance shown inFig. 1d. Algorithm A S provides more ideal phase performance and thereforecan provide an advantage over the construction procedure when toleranceto RF inhomogeneity is included in the optimization. The choice betweenalgorithms A and A S depends on the application, whether maximizing signalamplitude is more important than ideal phase performance. ◦ P P pulses
Two 180 ◦ P P pulses can be used to refocus transverse magnetization [17,18] by modifying the standard spin-echo sequence. For example, a 90 ◦ P P excitation pulse of length T p that produces a linear phase dispersion ∆ ωRT p (0 ≤ R ≤
1) as a function of offset ∆ ω could be followed by τ - 180 ◦ P P - ( τ + RT p ) - 180 ◦ P P . The first τ -delay already includes a phase evolutionequivalent to time RT p , which is included in the next delay of the standardspin-echo. The second inversion pulse then compensates the phase errors ofthe first [14], which is unnecessary if a single 180 ◦ UR is employed. A shortersequence for this example is ( τ − RT p ) - 180 ◦ P P - τ - 180 ◦ P P . The resultingmodified spin-echo procedures rotate magnetization 360 ◦ and are not URinversions, but this is not an issue for refocusing.BIP [14] and BIBOP [15, 16] pulses are optimized to provide exceptionalperformance as 180 ◦ P P pulses, which translates to similar performance whenincorporated into the above refocusing scheme. There is little room for im-proving the refocusing performance of two 180 ◦ P P pulses. A similar conclusionwas reached in the context of relatively high bandwidth selective pulses [46].On the other hand, a single 180 ◦ UR pulse can be simpler to incorporate intocomplex pulse sequences with respect to adjusting the timings and synchro-nization among various pulses. Conventional hard 180 ◦ pulses can be easilyreplaced in an existing sequence by 180 ◦ BURBOP pulses without further changeof the pulse sequence. By contrast, incorporating two 180 ◦ P P pulses can re-quire the adjustment of phases and a correspondingly detailed understandingof the pulse sequence. 8e therefore find at most a modest advantage in using algorithms A or A S to generate a 180 ◦ UR pulse compared to the overlapping spin-echo sequenceusing optimized 180 ◦ P P pulses. However, incorporating a time-dependent RFlimit into the optimal control algorithm does provide a distinct advantage forgenerating a single 180 ◦ UR pulse compared to existing pulses, as illustrated inwhat follows. T and A S,T
Broadband refocusing bandwidths of ∼
50 kHz, sufficient for high-field Cspectroscopy, are readily achieved using any of the pulse schemes discussedso far. The peak RF power required for the pulses is well-within hard-pulsepower limits for modern high resolution probes. However, in multipulse se-quences, repeated application of what might be deemed a modest power levelfor a single pulse can be a problem if the total energy delivered to the sample(integrated power) is too high. There are also limits on the total energythat can safely be delivered to a given probe. For these reasons, the mostgeneral and widespread applications impose peak power levels that are moreconservative than what might be necessary for broadband refocusing using atypical probe. We therefore incorporate a time-dependent RF limit into theoptimal control algorithms to keep peak power low for most of the pulse, butallow short increases in this limit where it can have the most benefit. Weutilize algorithms A T and A S,T (defined in § The best broadband refocusing performance available in the standardBruker pulse library satisfying the required conservative pulse power limits isobtained using the pulse designated Chirp80 [45]. This pulse is constructedfrom three adiabatic inversion pulses with pulse lengths in the ratio 1:2:1[19]. It utilizes for its shortest element a 500 ms smoothed chirp pulse [44]with 80 kHz sweep. The first 20% of the pulse rises smoothly to a maximumconstant RF amplitude of 11.26 kHz according to a sine function before de-creasing in the same fashion to zero during the final 20% of the pulse. Thefinal pulse is thus 2 ms long.Maintaining this pulse length and mindful of the given conservative peakRF amplitude, we designed the set of four pulses listed in Table 1. Formost of the pulse, the nominal RF amplitude is a constant 10 or 11 kHz. A9aximum RF amplitude of 15 kHz is applied for 60 µ s in the middle of thepulse, as illustrated in Fig. 2. This short increase in pulse amplitude providessignificant improvement in pulse performance compared to Chirp80.The amplitude profile shown in Fig. 2 is reminiscent of the hyperbolicsecant pulse [6], which maintains a low amplitude for most of the pulse with apeak in the middle. All four pulses show excellent performance over the listedranges in offset and RF field inhomogeneity/miscalibration. Performance iscomparable to the performance shown in Fig. 1 for higher power, constantamplitude pulses with nominal peak RF of 15 kHz. Pulse 4 provides themost relevant comparison, since it has a similar range of tolerance to RFinhomogeneity. As expected from the earlier results for algorithms A andA S , the best amplitude performance is obtained by algorithm A T and thebest phase performance by algorithm A S,T
Figure3 compares theoretical performance of pulses 1 and 4 from Table 1 tothe Chirp80 pulse. The new pulses significantly improve phase performanceover the targeted range of offsets and RF inhomgeneity/miscalibration. Ad-ditional quantitative comparison between pulse 4 and Chirp80 are providedin Figs.4 and 5, which also show the excellent agreement between simulationsand experimental pulse performance. Improvements in lineshape and phasethat are possible using the new pulses are shown in Fig. 6.
Pulse Algorithm RF nominal RF max Inhomogeneity Transformation Error(kHz) (kHz) optimization (amplitude) (phase)1 A T
11 15 ± < . < ◦ S,T
11 15 ± < . < . ◦ S,T
10 15 ± < . < . ◦ S,T
11 15 ± < < . ◦ Table 1: Four pulses optimized to execute a 180 ◦ universal rotation about the y-axis overresonance offsets of 50 kHz and RF field inhomogeneity listed in column 5. All pulses are2 ms long. For all but 60 µ s, pulse RF is constant at the value RF nominal , increasing toRF max in the middle of the pulse (see Fig. 2). Amplitude errors in the transformationare the maximum deviation from the target magnetization over the optimized ranges ofresonance offset and RF inhomogeneity, expressed as a percent. Similarly, phase errorsrepresent the maximum deviation from the target phase. Performance of pulses 1 and 4are shown in more detail in Fig. 3 . Experiment All experiments were implemented on a Bruker 750 MHz Avance III spec-trometer equipped with SGU units for RF control and linearized amplifiers,utilizing a triple-resonance PATXI probehead and gradients along the z -axis.Measurements are the residual HDO signal in a sample of 99.96% D O dopedwith CuSO to a T relaxation time of 100 ms at 298 ◦ K. Signals were obtainedat offsets between −
25 kHz to 25 kHz in steps of 500 Hz. To demonstrate thetolerance of the pulses to RF inhomogeneity/miscalibration, the experimentswere repeated with RF amplitude incremented by ±
15% and ±
25% relativeto the nominal maximum RF amplitude for each pulse (15 kHz for pulse 4of Table 1, 11.26 kHz for Chirp80). To reduce the effects of RF field inho-mogeneity within the coil itself, approximately 40 µ l of the sample solutionwas placed in a 5 mm Shigemi limited volume tube.
5. Conclusion
Acknowledgments
T.E.S. acknowledges support from the National Science Foundation underGrant CHE-0943441. B.L. thanks the Fonds der Chemischen Industrie andthe Deutsche Forschungsgemeinschaft (Emmy Noether fellowship LU 835/1-3) for support. S.J.G. acknowledges support from the DFG (GI 203/6-1),SFB 631 and the Fonds der Chemischen Industrie. The experiments wereperformed at the Bavarian NMR Center, Technische Universit¨at M¨unchen.11 . Appendix
The results of section 2.1.1 were derived as follows.
The rotation of a vector can be represented as the composition of three el-ementary (Euler) rotations about the fixed axes used to represent the vector:rotation by angle ψ about the z-axis followed by rotation θ about the x-axisand rotation ϕ about the z-axis. Using the notation c β = cos β, s β = sin β ,and R k ( β ) for a rotation by angle β about axis k , we have R z ( ψ ) = c ϕ − s ϕ s ϕ c ϕ
00 0 1 R x ( θ ) = c θ − s θ s θ c θ R z ( ϕ ) = c θ − s θ s θ c θ (10)giving R = R z ( ϕ ) R x ( θ ) R z ( ψ )= c ψ c ϕ − c θ s ψ s ϕ − c ϕ s ψ − c ψ c θ s ϕ s θ s ϕ c θ c ϕ s ψ + c ψ s ϕ c ψ c θ c ϕ − s ψ s ϕ − c ϕ s θ s ψ s θ c ψ s θ c θ = R R R R R R R R R (11)The Euler angles describing the rotation are thus θ = cos − [ R ] ϕ = tan − [ R / ( − R ) ] ψ = tan − [ R /R ] (12)Consider the rotation matrix W that produces the 90 ◦ P P transformation I z → − I y . For initial I z , the first z-rotation has no effect. Angle θ therefore12ives a direct comparison with the desired rotation angle of 90 ◦ , while ϕ givesthe phase deviation from the target state − I y . W also rotates initial states I x and I y to final states that can be arbitrary in the design of the originalPP transformation. This provides a great deal of flexibility in designing PPtransformations compared to UR transformations, which must rotate eachinitial state to a specific final state. For an ideal 90 ◦ UR rotation about thex-axis, ψ would be zero. It thus provides the orientations of the rotatedvectors W I x and W I y in the plane orthogonal to the final state W I z .The pulse that generates a 180 ◦ UR transformation from this particular 90 ◦ P P pulse is constructed from the phase-inverted pulse, denoted by 90
P P , followedby the time-reversed (tr) pulse 90 trP P [20]. At a resonance offset − ν , we havethe corresponding rotation operators W tr ( − ν ) = R z (180) W − ( ν ) R z (180)= W W − W W W − W − W − W W (13)and W ( − ν ) = R x (180) W ( ν ) R x (180)= W − W − W − W W W − W W W , (14)where R z (180) is zero except for ( − , − ,
1) along the diagonal, R x (180) issimilar, but with (1 , − , −
1) along the diagonal, and W − is the transposeof W .The 180 ◦ UR rotation matrix f W ( − ν ) = W tr ( − ν ) W ( − ν ) therefore differsfrom the identity matrix to the extent terms in the product for each matrixelement have different signs. For example, f W is equal to W − W + W rather than W + W + W = 1. Adding and subtracting W in theexpression for f W gives 1 − W . Proceding similarly, all the matrix elementscan be written in terms of their difference from zero (off-diagonal) or ± f W = − W W W W W − W W − W W W W W − W W − W + W ) (15)13or the 180 ◦ UR rotation operator f W at resonance offset − ν in terms of elements W ij of the 90 ◦ P P rotation operator at offset ν . Differences in performanceat offsets ± ν are small to the extent the PP optimization is successful ingenerating uniform performance over a symmetric range of resonance offsets.The Euler angles for f W obtained using Eq. [ 12 ] give ˜ ϕ = ˜ ψ . This resultprovides geometric insight into the performance of f W . Rotation about the z -axis followed by 180 ◦ rotation about x then rotation by the original amountabout z will return x to x and send y and z to − y and − z , respectively. Thedeviation from the ideal goal of a 180 ◦ rotation determines the errors in the180 ◦ UR transformation. For initial I x , the phase angle φ for the rotated state f W I x relative to the target state I x is tan φ = f W / f W . Using Eq. [ 11 ] andadapting our earlier notation to ˜ c β = cos ˜ β , etc., givestan φ = ˜ c ψ ˜ s ψ (1 + ˜ c θ )˜ c ψ (1 − ˜ c θ tan ˜ ψ )= tan ˜ ψ c θ − ˜ c θ tan ˜ ψ (16)The extrema for tan φ as a function of ˜ ψ occur for d (tan φ ) /d ˜ ψ = 0. Thereare two solutions, one of which is an inflection point and the other producinga maximum for tan ˜ ψ = − ˜ c − / θ , so thattan φ max = 1 + ˜ c θ √− ˜ c θ (17)According to Eq. [ 12 ] and Eq. [ 15 ],cos ˜ θ = f W = − W + W )= − s θ s ϕ + c θ )= − c δθ s ϕ + s δθ ) , (18)where we have substituted θ = π/ δθ .Both δθ and ϕ are small for a good 90 ◦ P P pulse. Expanding c δθ ≈ − ( δθ ) / s ϕ ≈ ϕ , s δθ ≈ δθ , performing the multiplications in Eq. [ 18 ], andkeeping terms to second order gives, with ˜ θ = π + δ ˜ θ ,cos ˜ θ ≈ − δθ + ϕ )14os δ ˜ θ ≈ − δθ + ϕ )1 − δ ˜ θ / ≈ − δθ + ϕ ) δ ˜ θ ≈ p δθ + ϕ . (19)Thus, the deviation δ ˜ θ from the ideal 180 ◦ UR rotation angle is slightly morethan twice the deviation δθ from the ideal 90 ◦ P P rotation angle.Using Eq. [ 19 ] for cos ˜ θ , the small-angle approximation for tan φ , andapplying (1 − x ) / ≈ − x/ x to tan ˜ ψ = 1 / ( − cos ˜ θ ) / givestan φ max ≈ φ max . δθ + ϕ − ( δθ + ϕ ) ≈ δθ + ϕ (20)Similarly, for initial I y , the phase relative to the target − y -axis is tan φ = f W / f W , which results in tan ˜ ψ being replaced by cot ˜ ψ in Eq. [ 16 ]. Thesolution for tan φ max then occurs for cot ˜ ψ = ( − cos ˜ θ ) − / , giving the samebound for φ max as in Eq. [ 20 ]. Alternatively, we can describe a rotation in terms of rotation angle η aboutan axis defined by unit vector ˆ n to obtain η = cos − Tr R − n x = R − R ηn y = R − R ηn z = R − R η. (21)Applying this to f W in Eq. [ 15 ] shows n y = 0, as expected from symmetryarguments noted in section 2.1.1. The rotation axis ˆ n makes an angle α withrespect to the x -axis given by tan α = n z /n x = ( f W − f W ) / ( f W − f W ).Following arguments similar to those leading to Eq. [ 16 ] gives, for small α , α ≈ tan α = 2˜ c ψ ˜ s ψ (1 + ˜ c θ )2˜ c ψ ˜ s θ ≤ (1 + ˜ c θ ) / ˜ s θ ≈ p δθ + φ . (22)15ere, we have written ˜ s θ = (1 − ˜ c θ ) / , substituted from Eq. [ 19 ] keepingterms to second order, and used the maximum value of one for ˜ s ψ .We obtain similarly from Eq. [ 21 ] cos η = − W = f W after rear-ranging terms as in the discussion preceding Eq. [ 15 ]. Substituting as abovegives f W = ˜ c ψ ˜ c θ − ˜ s ψ . Writing ˜ s ψ = 1 − ˜ c ψ and employing the maximumvalue of one for ˜ c ψ gives cos η ≤ cos ˜ θ (23)and the deviation of each angle from the ideal rotation of 180 ◦ can be rela-tively quantified as δη ≤ δ ˜ θ, (24)with δ ˜ θ given in terms of the 90 ◦ P P angles in Eq. [ 19 ].
References [1] M. H. Levitt and R. Freeman, NMR population inversion using a com-posite pulse, J. Magn. Reson. 33 (1979) 473–476.[2] A. J. Shaka, J. Keeler, T. Frenkiel, and R. Freeman, An improved se-quence for broadband decoupling:WALTZ-16, J. Magn. Reson. 52, 335338 (1983).[3] A. J. Shaka, J. Keeler, and R. Freeman, Evaluation of a new broadbanddecoupling sequence: WALTZ-16, J. Magn. Reson. 53, 313340 (1983).[4] J. Baum, R. Tycko, and A. Pines, Broadband population inversion byphase modulated pulses, J. Chem. Phys. 79, 46434644 (1983).[5] M. H. Levitt and R. R. Ernst, Composite pulses constructed by a recur-sive expansion procedure, J. Magn. Reson. 55, 247254 (1983).[6] M. S. Silver, R. I. Joseph, and D. I. Hoult, Selective spin inversion in nu-clear magnetic resonance and coherence optics through an exact solutionof the Bloch-Riccati equation, Phys. Rev. A. 31, 27532755 (1985).[7] J. Baum, R. Tycko, and A. Pines, Broadband and adiabatic inversionof a two-level system by phase-modulated pulses, Phys. Revs. A 32,34353446 (1985).[8] A. J. Shaka, Composite pulses for ultra-broadband spin inversion, Chem.Phys. Lett. 120, 201205 (1985). 169] M. H. Levitt, Composite pulses, Prog. NMR Spectrosc. 18 (1986) 61–122.[10] E. Kupce and R. Freeman, Adiabatic pulses for wideband inversion andbroadband decoupling, J. Magn. Reson. A 115, 273276 (1995).[11] E. Kupce and R. Freeman, Optimized adiabatic pulses for widebandspin inversion, J. Magn. Reson. A 118, 299303 (1996).[12] A. Tannus and M. Garwood, Improved performance of frequency-sweptpulses using offset-independent adiabaticity, J. Magn. Reson. A 120, 133137 (1996).[13] T. Hwang, P. van Zijl, and M. Garwood, Fast broadband inversion byadiabatic pulses, J. Magn. Reson. 133, 200203 (1998).[14] M. A. Smith, H. Hu, A. J. Shaka, Improved broadband inversion per-formance for NMR in liquids, J. Magn. Reson. 151 (2001) 269–283.[15] K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser, B. Luy, Exploringthe limits of broadband excitation and inversion pulses, J. Magn. Reson.170 (2004) 236–243.[16] K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser, B. Luy, Exploringthe limits of broadband excitation and inversion: II. Rf-power optimizedpulses, J. Magn. Reson. 194 (2008) 58–66.[17] M. H. Levitt and R. Freeman, Compensation for pulse imperfections inNMR spin-echo experiments, J. Magn. Reson. 43 (1980) 65–80.[18] S. Conolly, G. Glover, D. Nishimura, and A. Macovski, A reduced powerselective adiabatic spin-echo pulse sequence, Magn. Reson. Med. 18(1991) 2838.[19] T.L. Hwang, PCM Van Zij, M. Garwood. Broadband Adiabatic Refo-cusing without Phase Distortion. J. Magn. Reson., 124 (1997) 250–254.[20] B. Luy, K. Kobzar, T.E. Skinner, N. Khaneja, S.J. Glaser, Constructionof Universal Rotations from Point to Point Transformations, J. Magn.Reson. 176 (2005) 179–186. 1721] A.J.Shaka and A.Pines, Symmetric phase-alternating composite pulses,J. Magn. Reson. 71 (1987) 495–503.[22] S. Conolly, D. Nishimura, A. Macovski, Optimal control solutions tothe magnetic resonance selective excitation problem, IEEE Trans. Med.Imaging MI-5 (1986) 106–115.[23] J. Mao, T.H. Mareci, K.N. Scott, E.R. Andrew, Selective inversion ra-diofrequency pulses by optimal control, J. Magn. Reson. 70 (1986)310–318.[24] V. Smith, J. Kurhanewicz, and T. L. James, Solvent-suppression pulses.I. Design using optimal control theory, J. Magn. Reson. 95 (1991)41–60.[25] V. Smith, J. Kurhanewicz, and T. L. James, Solvent-suppressionpulses. II. In vitro and in vivo testing of optimal-control-theory pulses,J. Magn. Reson. 95 (1991) 61–70.[26] D. Rosenfeld, Y. Zur, Design of adiabatic selective pulses using optimalcontrol theory, Magn. Reson. Med. 36 (1996) 401–409.[27] T.E. Skinner, T.O. Reiss, B. Luy, N. Khaneja, S.J. Glaser, Applicationof optimal control theory to the design of broadband excitation pulsesfor high resolution NMR, J. Magn. Reson. 163 (2003) 8–15.[28] T.E. Skinner, T.O. Reiss, B. Luy, N. Khaneja, S.J. Glaser, Reducingthe duration of broadband excitation pulses using optimal control withlimited RF amplitude, J. Magn. Reson. 167 (2004) 68–74.[29] T.E. Skinner, T.O. Reiss, B. Luy, N. Khaneja, S.J. Glaser, Tailoring theoptimal control cost function to a desired output: application to mini-mizing phase errors in short broadband excitation pulses, J. Magn. Re-son. 172 (2005) 17–23.[30] T.E. Skinner, K. Kobzar, B. Luy, R. Bendall, W. Bermel, N. Khaneja,S.J. Glaser, Optimal control design of constant amplitude phase-modulated pulses: application to calibration-free broadband excitation,J. Magn. Reson. 179 (2006) 241–249.1831] N. I. Gershenzon, K. Kobzar, B. Luy, S. J. Glaser, T. E. Skinner, Opti-mal control design of excitation pulses that accommodate relaxation, J.Magn. Reson. 188 (2007) 330–336.[32] N. I. Gershenzon, T. E. Skinner, B. Brutscher, N. Khaneja, M. Nim-balkar, B. Luy, S. J. Glaser, Linear phase slope in pulse design: Appli-cation to coherence transfer, J. Magn. Reson. 192 (2008) 235–243.[33] N.I. Gershenzon, D.F. Miller, T.E. Skinner, The design of excitationpulses for spin systems using optimal control theory:With applicationto NMR spectroscopy, Optim. Control Appl. Meth. 30 (2009) 463–475.[34] T.E. Skinner, N.I. Gershenzon, Optimal control design of pulse shapesas analytic functions, J. Magn. Reson. 204 (2010) 248–55.[35] T. E. Skinner, M. Braun, K. Woelk, N. I. Gershenzon, S. J. Glaser,Design and application of robust rf pulses for toroid cavity NMR spec-troscopy, J. Magn. Reson. 209 (2011) 282–290.[36] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨uggen, S. J. Glaser,Optimal control of coupled spin dynamics: design of NMR pulse se-quences by gradient ascent algorithms, J. Magn. Reson. 172 (2005)296–305.[37] K. Kobzar, PhD Thesis, Technische Universit¨at M¨unchen, 2007.[38] T. W. Borneman, M. D. H¨urlimann, D. G. Cory, Application of optimalcontrol to CPMG refocusing pulse design, J. Magn. Reson. 207 (2010)220-233.[39] C. Rangan and P.H. Bucksbaum, Optimally shaped terahertz pulses forphase retrieval in a Rydberg-atom data register, Phys. Rev. A 64 (2001)033417-5 .[40] J.P. Palao and R. Kosloff, Quantum computing by an optimal control al-gorithm for unitary transformations, Phys. Rev. Lett. 89 (2002) 188301-4.[41] T. Schulte-Herbr¨uggen, A. Sp¨orl, N. Khaneja, S. J. Glaser, Optimalcontrol-based efficient synthesis of building blocks of quantum algo-rithms seen in perspective from network complexity towards time com-plexity, Phys. Rev. A 72 (2005) 042331-7.1942] R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth,J. Appl. Phys. 28 (1957) 49–52.[43] J.T. Ngo, P.G. Morris, NMR pulse symmetry, J. Magn. Reson. 74 (1987)122–133.[44] J.-M. B¨ohlen and G. Bodenhausen, Experimental aspects of chirp NMRspectroscopy, J. Magn. Reson. Series A 102 (1993) 293–301.[45] M. Kock, R. Kerssebaum and W. Bermel, A broadband ADEQUATEpulse sequence using chirp pulses. Magn. Reson. Chem. 2003; 41: 65–69.[46] G.B. Matson, K. Young, L.G. Kaiser, RF pulses for in vivo spectroscopyat high field designed under conditions of limited power using optimalcontrol, J. Magn. Reson. 199 (2009) 30–40.20 z → − M z (Transfer Efficiency) B / B M x → − M x (Transfer Efficiency) M x → − M x (Phase deviation) B / B B / B B / B Offset (kHz) −20 0 200.711.3
Offset (kHz) −20 0 20
Offset (kHz) −20 0 200.98 0.3 a)b)c)d) Figure 1: Theoretical performance of four 180 ◦ UR pulses for inversion of magnetizationabout the y-axis, designed using different algorithms. The first two columns show thetransfer efficiency for the labeled transformations relative to ideal or complete transfer.The last column displays phase deviation in degrees relative to the labeled target state.The nominal peak RF amplitude for all the pulses is B = 15 kHz, optimized to performover a resonance offset range of 50 kHz and variation in RF homogeneity/calibration of ± a) constructed from the 1 ms 90 ◦ P P pulse of Ref. [30] preceded by its time- and phase-reversed waveform [20], pulse length T p = 2 ms. b) algorithm A, T p = 2 ms. c) algorithm A, T p = 1 ms. d) algorithm A S ,which incorporates the symmetry principle used in a), T p = 2 ms. A m p li t ude ( k H z ) Time (ms) P ha s e ( deg r ee ) Figure 2: The amplitude and phase of 180 ◦ UR pulse 4 from Table 1 obtained using algorithmA S,T . A conservative limit of 11 kHz for the peak RF applied for 2 ms is relaxed to allow asafe peak of 15 kHz for 60 µ s. The pulse is amplitude-symmetric and phase-antisymmetricin time, incorporating the symmetry of the construction procedure from Ref. [20]. igure 3: Similar to Fig. 1, but the maximum RF amplitude is allowed to float for asufficiently short time period, giving the reduced-power pulse in Fig. 2. All three pulsesare designed to operate over a resonance offset of 50 kHz and a range of variation in RFhomogeneity/calibration relative to the ideal B , as given in Table 1. a) pulse 1 of Table1, RF tolerance ± b) pulse 4 of Table 1, RF tolerance ± c) composite adiabaticrefocusing [19] using pulse Chirp80 from the Bruker pulse library [45]. F scaling-25 %-15 %0 %15 %25 % -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] Figure 4: Further quantitative detail for the M x → − M x transformation from Fig. 3b forpulse 4 of Table 1 (black) and Fig. 3c for Chirp80 (red), plotted for RF scalings of ± ±
25% relative to the nominal maximum RF amplitude at 0% (15 kHz, pulse 4 and11.26 kHz, Chirp80). Theoretical values for the inversion profile are plotted on the left asa function of resonance offset, with phase deviation ϕ relative to the target − M x plottedon the right. Adiabatic Chirp80 produces significant phase errors within the bandwidthfor all RF scalings, in contrast to the almost ideal performance of the optimal control180 ◦ BURBOP pulse. F scaling-25 %-15 %0 %15 %25 % -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] -20 -10 0 10 20-1-0.95-0.9 n [kHz] M x -20 -10 0 10 20-20-10010 n [kHz] φ [ deg ] Figure 5: Experimental measurements of the inversion profile (left) and phase deviation(right) corresponding to the simulations in Fig. 4, showing excellent agreement betweenthe experimental and theoretical performance of the pulses. diabaticOptimal control -25 % -15 % 0 % 15 % 25 %RF scaling Figure 6: Experimental lineshapes comparing optimal control 180 ◦ UR pulse 4 of Table 1and adiabatic Chirp80 at a resonance offset of −
25 kHz for RF scalings of ±
15% and ±
25% relative to the nominal maximum RF amplitude of 15 kHz at 0%. Simulations (notshown) are in excellent agreement with the experiments, as already shown in Figures 4and 5. Chirp80 produces significant experimental phase errors of 20 ◦ , 15 ◦ , 9 ◦ , 11 ◦ , and5 ◦ (reading left-to-right across the RF scalings in the figure) in contrast to the experimentalperformance of 0.7 ◦ , 0.7 ◦ , 0.9 ◦ , 2.6 ◦ , 2.5 ◦ for the optimal control 180 ◦ BURBOP pulse.pulse.