Elsevier

Journal of Magnetic Resonance

Volume 225, December 2012, Pages 142-160
Journal of Magnetic Resonance

Exploring the limits of broadband 90° and 180° universal rotation pulses

https://doi.org/10.1016/j.jmr.2012.09.013Get rights and content

Abstract

90° and 180° universal rotation (UR) pulses are two of the most important classes of pulses in modern NMR spectroscopy. This article presents a systematic study characterizing the achievable performance of these pulses as functions of bandwidth, pulse length, and tolerance to B1-field inhomogeneity/miscalibration. After an evaluation of different quality factors employed in pulse design algorithms based on optimal control theory, resulting pulses are discussed in detail with a special focus on pulse symmetry. The vast majority of resulting BURBOP (broadband universal rotations by optimal control) pulses are either fully symmetric or have one symmetric and one antisymmetric Cartesian rf component, where the importance of the first symmetry has not been demonstrated yet and the latter one matches the symmetry that results from a previously derived construction principle of universal rotation pulses out of point-to-point pulses [3]. Optimized BURBOP pulses are shown to perform better than previously reported UR pulses, resulting in shorter pulse durations for the same quality of broadband rotations. From a comparison of qualities of effective universal rotations, we find that the application of a single optimal refocusing pulse matches or improves the performance of two consecutive inversion pulses in INEPT-like pulse sequence elements of the same total duration.

Highlights

► A systematic study of performances of universal 90° and 180° pulses is presented. ► Pulses were studied with respect to bandwidth, pulse length, tolerance to B1-inhomogeneity. ► The majority of time-optimal pulses belongs to one of two specific symmetry classes. ► BURBOP pulses found are shown to perform better than previously published pulses. ► UR 180° pulses perform better than two consecutive PP inversion pulses in INEPT steps.

Introduction

The physical limits of pulse performance are of fundamental and practical importance for NMR spectroscopy. A thorough characterization of these limits allows one to choose the best pulse for a given application.

Currently, hard 90° and 180° pulses are the workhorses of modern NMR spectroscopy. They are commonly used in the majority of pulse sequences, but the development of spectrometers with magnetic field strengths on the order of 1 GHz and the large chemical shift ranges of nuclei like 13C, 19F, or 31P pose serious challenges to their utility. A simple hard pulse provides effective excitation or inversion only over a limited range of resonance offsets that cannot be increased significantly due to pulse power constraints.

For the many applications requiring uniform excitation or inversion over resonance offsets that are greater than the maximum available B1 field, computer-optimized pulse shapes can provide an effective broadband solution. They can also be designed with a defined tolerance to B1 field inhomogeneity.

Such pulses typically belong to the class of point-to-point (PP) pulses (also called class B2 pulses [1]) designed to transfer a single initial magnetization component, for example Iz, to a specific target component, e.g. Ix for excitation and -Iz for inversion. Whenever the transfer of a single magnetization component is desirable in a pulse sequence, broadband PP pulses will be the adequate replacement for hard pulses.

In many situations, however, more than one magnetization component must be transformed via a universal rotation (UR) capable of rotating any orientation of the initial magnetization about the same axis.

UR pulses (also denoted as class A pulses [1], constant rotation pulses [2], general rotation pulses, plane rotation pulses, or simply universal pulses) are characterized by a well-defined rotation axis and rotation angle. Initial vectors pointing along the x, y and z direction are rotated by UR pulses to well-defined new orientations, e.g. along the -z, y and x direction, respectively, for a UR (90y°) pulse (c.f. Fig. 1a). In contrast, PP pulses are only required to rotate a single initial vector (e.g. pointing along the z axis) to a desired final orientation, e.g. along the x axis for a PP(zx) pulse (c.f. Fig. 1b). In this example, initial x and y magnetization vectors are allowed to end up anywhere in the yz plane. Each PP pulse can be realized by an infinite set of different combinations of rotation axes and rotation angles [3]. For example, a PP(zx) pulse can be realized using arbitrary rotation angles between 90° (with rotation axis (0, 1, 0), i.e. the y axis) and 180° (with rotation axis (1/2,0,1/2), corresponding to the bisecting line of the angle between the x and z axis). As UR pulses transform all three axes in a controlled way while only one axis is controlled in PP pulses, UR pulses can always replace corresponding PP pulses in a pulse sequence but not vice versa.

We have previously characterized the performance limits of broadband PP excitation and inversion pulses [4], [5]. In addition, direct UR optimizations have provided UR pulse shapes with exceptional performance for specific problems [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Pulse sequences in need of 90° UR pulses include the mixing step in COSY [16], ADEQUATE [17], and INADEQUATE-type [18], [19], [20] sequences, or the sensitivity enhancement [21], [22], [23] in HSQC-type experiments. The crucial refocusing of chemical shift evolution requires a 180° UR pulse. Ideally, this refocusing pulse should provide the same effective universal rotation for a given bandwidth as an on-resonant hard 180° pulse.

In this article, we extend our work on optimal pulse performance and consider the physical bounds of broadband 90° and 180° UR pulses for the first time. A major optimization effort begun years ago [8] has been applied to the de novo design of UR pulses to yield important new results, giving the best UR pulse performance as functions of pulse length, bandwidth, and B1-tolerance using the GRAPE algorithm [6], one of the most efficient optimization tools currently known to the authors.

Although UR pulse design is generally considered to be more difficult than optimizing PP pulses, traditional pulse design methods have yielded a number of composite or shaped UR pulses in the last decades [1], [2], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. More recently, a principle has been derived which allows the construction of a UR pulse with flip angle α based on PP pulses with effective flip angle α/2 starting with initial magnetization along the y-axis [3]. The open question is the degree to which existing UR pulses achieve the best possible performance. Given the crucial role of UR pulses in the wide variety of important coherence transfer applications in NMR, a thorough characterization of the performance limits of UR pulses is highly desirable.

After the introduction of a general pulse nomenclature, we introduce and evaluate the convergence criteria of two important classes of cost function. The evaluation of these classes turns out to be of utmost importance for the vast number of individual optimizations performed thereafter and will be of general interest for future optimization protocols. We then provide time-optimal curves for various bandwidths and tolerances to rf-scaling and discuss and classify the large number of optimal pulse shapes obtained. Different symmetries emerge among the pulse shapes, which are optimized without any symmetry constraints. Most noteably, one symmetry relates to the aforementioned construction principle [3]. The performance of these classes of symmetric pulse shapes are characterized and tested with respect to time-optimality. Finally, optimal UR pulse performance is compared to previously published UR pulses, showing the significant performance improvements revealed by the present study. We also determine whether a single UR 180° pulse can provide superior performance in INEPT-type transfer elements compared to the well-known use of two PP inversion pulses with suitable delays [40], [41]. Mathematical derivations of symmetry and cost function relations used in the main text are provided in an appendix.

Section snippets

Pulse nomenclature

As with BEBOP [4], [42], [43], [44], [45], BIBOP [4], power-BEBOP [5], power-BIBOP [5], RC-BEBOP [46] and ICEBERG [47] pulses, the universal rotation pulses presented in this article describe a whole family of pulses with transfer properties close to the physical limits for a given set of rf amplitude, bandwidth, and time-constraints. For the accurate description of corresponding pulses we therefore would like to introduce a common nomenclature for BURBOP (Broadband Universal Rotations By

Performance functions and GRAPE algorithm for UR-pulses

Optimal control theory is a mathematical field that is concerned with control policies that can be deduced using optimization algorithms [48], [49]. In other words, this theory provides a way to find such controls for a given dynamical system, which drive it from an initial state to a target state in a most efficient way. It is based on a classical Euler–Lagrange formalism, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s.

Instead of optimizing a point-to-point problem with

Phase factors and convergence of the algorithm

Although, as noted in the previous section, the global phase of the propagator representing a given pulse has no effect on the final state ρ(t), global phase factors do play an important role in the optimization of such unitary operators [7], [8], [10]. As discussed in Appendix A.2, for a single spin 1/2, only two global phase factors are possible: ei0=+1 or eiπ=-1. Optimizations based on the performance function Φ1 as defined in Eq. (8) do not distinguish between these two possibilities for

Limits of 90° and 180° universal rotation pulses with respect to offset bandwidth

A systematic study of pulse performance, similar to the ones performed previously for PP excitation pulses [4], [5], was performed for UR pulses using the described algorithm based on optimal control theory. Sets of 90° and 180° rotations were calculated for different bandwidths ranging from 10 kHz to 50 kHz with the rf amplitude limited to 10 kHz in all cases. Optimizations were performed using the performance function Φ¯0 (Eq. (5)) with the two target propagators ±UF for the case of 90° UR

Directly optimized UR pulses vs. constructed UR pulses

Considering the high degree of symmetry of most optimized BURBOP pulses, it is an interesting question as to how far such symmetric pulses represent the global optimum. Since broadband UR pulses can be constructed from symmetry principles out of two PP pulses with half the effective flip angle [3], we consider the relative merits of directly optimized UR pulses compared to constructed pulses, as in [14] for 180° UR pulses, before looking into more general symmetry principles.

For a comparison,

Symmetry properties of universal rotation pulses

For a systematic examination of the symmetries found empirically in the optimized UR pulses, we consider the following four types of pulse symmetries:

  • the fully symmetric case, with both x and y component symmetric with respect to the center of the pulse (also referred to as type I pulse or xS,yS),

  • antisymmetric x and symmetric y component (type II pulse or xAS,yS),

  • symmetric x and antisymmetric y component (type III pulse or xS,yAS),

  • and the fully antisymmetric case (type IV pulse or xAS,yAS).

B1 inhomogeneity

UR and PP pulses in this article were optimized so far without considering variations in rf amplitude as, for example, imposed by B1 inhomogeneities in a real experimental setup. For a systematic study of the effect of rf-variations and for best comparability with previous results on excitation and inversion pulses [4], [5], we focused on optimizations for a fixed bandwidth of 20 kHz with varying constraints on the compensation of rf-variations expressed by the relative range of allowed rf

Discussion

Modern high resolution NMR spectroscopy with magnetic field strengths of up to 1 GHz results in large offset ranges that have to be uniformly covered in corresponding pulse sequences. Especially for frequently used heteronuclei like 13C, 19F, 15N, and 31P, their large chemical shift ranges pose serious problems for conventional probeheads using hard pulses. High frequencies, in addition, restrict the available maximum rf amplitudes due to technological limitation. As a result, the use of phase-

Conclusion

In this article, a systematic study of bounds for the physical performance limits of broadband 90° and 180° universal rotation pulses with respect to pulse length, bandwidth, and compensation for rf-variations is presented, which can be regarded as an extension to our previous work on point-to-point pulses [4], [5]. The study is based on optimizations using principles of optimal control.

The main results are the achievable qualities of broadband 90° and 180° UR pulses for a large variety of

Acknowledgments

T.E.S. acknowledges support from the National Science Foundation under Grant CHE-0943441. S.J.G. acknowledges support from the DFG (GL 203/6-1), SFB 631 and the Fonds der Chemischen Industrie. B.L. thanks the DFG (Heisenberg Program LU 835/2,3,4,7 and Forschergruppe FOR 934) and the Fonds der Chemischen Industrie.

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