Tailoring the optimal control cost function to a desired output: application to minimizing phase errors in short broadband excitation pulses

Band-selective universal 90° and 180° rotation pulses covering the aliphatic carbon chemical shift range for triple resonance experiments on 1.2 GHz spectrometers

Biomolecular NMR spectroscopy requires large magnetic field strengths for high spectral resolution. Today's highest fields comprise proton Larmor frequencies of 1.2 GHz and even larger field strengths are to be expected in the future. In protein triple resonance experiments, various carbon bandwidths need to be excited by selective pulses including the large aliphatic chemical shift range. When the spectrometer field strength is increased, the length of these pulses has to be decreased by the same factor, resulting in higher rf-amplitudes being necessary in order to cover the required frequency region. Currently available band-selective pulses like Q3/Q5 excite a narrow bandwidth compared to the necessary rf-amplitude. Because the maximum rf-power allowed in probeheads is limited, none of the selective universal rotation pulses reported so far is able to cover the full [Formula: see text]C aliphatic region on 1.2 GHz spectrometers. In this work, we present band-selective 90° and 180° universal rotation pulses (SURBOP90 and SURBOP180) that have a higher ratio of selective bandwidth to maximum rf-amplitude than standard pulses. Simulations show that these pulses perform better than standard pulses, e. g. Q3/Q5, especially when rf-inhomogeneity is taken into account. The theoretical and experimental performance is demonstrated in offset profiles and by implementing the SURBOP pulses in an HNCACB experiment at 1.2 GHz.

SORDOR pulses: expansion of the Böhlen-Bodenhausen scheme for low-power broadband magnetic resonance

A novel type of efficient broadband pulse, called second-order phase dispersion by optimised rotation (SORDOR), has recently been introduced. In contrast to adiabatic excitation, SORDOR-90 pulses provide effective transverse 90∘ rotations throughout their bandwidth, with a quadratic offset dependence of the phase in the x , y plane. Together with phase-matched SORDOR-180 pulses, this enables the Böhlen-Bodenhausen broadband refocusing approach for linearly frequency-swept pulses to be extended to any type of 90∘ /180∘ pulse-delay sequence. Example pulse shapes are characterised in theory and experiment, and an example application is given with a 19 F -PROJECT experiment for measuring relaxation times with reduced distortions due to J -coupling evolution.

Concurrent J-evolving refocusing pulses

Conventional refocusing pulses are optimised for a single spin without considering any type of coupling. However, despite the fact that most couplings will result in undesired distortions, refocusing in delay-pulse-delay-type sequences with desired heteronuclear coherence transfer might be enhanced considerably by including coupling evolution into pulse design. We provide a proof of principle study for a Hydrogen-Carbon refocusing pulse sandwich with inherent J-evolution following the previously reported ICEBERG-principle with improved performance in terms of refocusing performance and/or overall effective coherence transfer time. Pulses are optimised using optimal control theory with a newly derived quality factor and z-controls as an efficient tool to speed up calculations. Pulses are characterised in theory and experiment and compared to conventional concurrent refocusing pulses, clearly showing an improvement for the J-evolving pulse sandwich. As a side-product, also efficient J-compensated resfocusing pulse sandwiches - termed BUBU pulses following the nomenclature of previous J-compensated BUBI and BEBE^tr pulse sandwiches - have been optimised.

Chirped ordered pulses for ultra-broadband ESR spectroscopy

Recently, applications of swept-frequency pulses proved to be a useful approach to circumvent the problem of limited excitation bandwidth in pulsed ESR posed by conventional pulses. Here, we present a chirped excitation sequence, CHirped ORdered pulses for Ultra-broadband Spectroscopy (CHORUS), for ultra-broadband ESR spectroscopy. It will be demonstrated that the application of this sequence can address the problems of excitation non-uniformity and sensitivity to instrumental instabilities to a greater extent compared to the current state of the art. This sequence is highly promising for finding applications beyond single excitation in many ESR experiments. Theoretical and experimental results for the proposed method are presented along with calibration strategies for experimental implementation.

Improved ultra-broadband chirp excitation

The design and application of ultra-broadband excitation pulses have been among the most interesting and timely areas in NMR and EPR methodology in recent years, due especially to advances in hardware design in EPR, the advent and popularity of high- and ultrahigh-field NMR, and the application of numerical methods like optimal control theory to the design and optimization of radiofrequency pulses and pulse sequences. In this communication, we present a short, robust, and flexible version of the CHORUS family of constant-phase, very broadband excitation sequences. We demonstrate that more than 0.5 MHz excitation with uniform amplitudes and phases can be achieved with this excitation sequence.

ASAP-HSQC-TOCSY for fast spin system identification and extraction of long-range couplings

Based on Ernst-angle-type excitation and Acceleration by Sharing Adjacent Polarization (ASAP), a fast HSQC-TOCSY experiment is introduced. In the approach, the DIPSI-2 isotropic mixing period of the ASAP-HSQC is simply shifted, which provides a TOCSY period without additional application of rf-energy. The ASAP-HSQC-TOCSY allows the acquisition of a conventional 2D in about 30 s. Alternatively, it allows the acquisition of highly carbon-resolved spectra (several Hz digital resolution) on the order of minutes. An ASAP-HSQC-TOCSY-IPAP variant, finally, allows the sign-sensitive extraction of heteronuclear long-range coupling constants from a pair of highly resolved spectra in less than an hour. Pulse sequences, several example spectra, and a discussion of results are given.

Using optimal control methods with constraints to generate singlet states in NMR

A method is proposed for optimizing the performance of the APSOC (Adiabatic-Passage Spin Order Conversion) technique, which can be exploited in NMR experiments with singlet spin states. In this technique magnetization-to-singlet conversion (and singlet-to-magnetization conversion) is performed by using adiabatically ramped RF-fields. Optimization utilizes the GRAPE (Gradient Ascent Pulse Engineering) approach, in which for a fixed search area we assume monotonicity to the envelope of the RF-field. Such an approach allows one to achieve much better performance for APSOC; consequently, the efficiency of magnetization-to-singlet conversion is greatly improved as compared to simple model RF-ramps, e.g., linear ramps. We also demonstrate that the optimization method is reasonably robust to possible inaccuracies in determining NMR parameters of the spin system under study and also in setting the RF-field parameters. The present approach can be exploited in other NMR and EPR applications using adiabatic switching of spin Hamiltonians.

Cooperative broadband spin echoes through optimal control

The Hahn echo sequence is one of the most common building blocks in magnetic resonance, consisting of an excitation pulse and a refocusing pulse. Conventional approaches to improve the performance of echo experiments focused on the optimization of individual pulses, compensating their own imperfections. Here we present an approach to concurrently design both pulses such that they also compensate each others imperfections. The fact that for such cooperative pulses the individual pulses do not need to be perfect provides additional degrees of freedom, resulting in improved overall Hahn echo performance. Single-scan cooperative pulses are compared to conventional approaches by simulations as well as experiments.

Broadband RF-amplitude-dependent flip angle pulses with linear phase slope

Pulse sequences in NMR spectroscopy sometimes require the application of pulses with effective flip angles different from 90° and 180°. Previously (Magn. Reson. Chem. 2015, 53, 886-893), offset-compensated broadband excitation pulses with RF-amplitude-dependent effective flip angles (RADFA) were introduced that are applicable in such cases. However, especially RF-amplitude-restricted RADFA pulses turned out to perform not as good as desired in terms of achievable bandwidths. Here, a class of RF-amplitude-restricted RADFA pulses with linear phase slope is introduced that allows excitation over much larger bandwidths with better performance. In this theoretical work, the basic principle of the pulse class is explained, their physical limits explored, and their properties, also compared with other pulse classes, discussed in detail. Copyright © 2017 John Wiley & Sons, Ltd.

Improvements, extensions, and practical aspects of rapid ASAP-HSQC and ALSOFAST-HSQC pulse sequences for studying small molecules at natural abundance

Previously we introduced two novel NMR experiments for small molecules, the so-called ASAP-HSQC and ALSOFAST-HSQC (Schulze-Sünninghausen et al., 2014), which allow the detection of heteronuclear one-bond correlations in less than 30s at natural abundance. We propose an improved symmetrized pulse scheme of the basic experiment to minimize artifact intensities and the combination with non-uniform sampling to enable the acquisition of conventional HSQC spectra in as short as a couple of seconds and extremely ¹³C-resolved spectra in less than ten minutes. Based on steady state investigations, a first estimate to relative achievable signal intensities with respect to conventional, ASAP-, and ALSOFAST-HSQC experiments is given. In addition, we describe several extensions to the basic pulse schemes, like a multiplicity-edited version, a revised symmetrized CLIP-ASAP-HSQC, an ASAP-/ALSOFAST-HSQC sequence with broadband BIRD-based ¹H,¹H decoupling, and a symmetrized sequence optimized for water suppression. Finally, RF-power considerations with respect to the high duty cycle of the experiments are given.

Perspectives of shaped pulses for EPR spectroscopy

This article describes current uses of shaped pulses, generated by an arbitrary waveform generator, in the field of EPR spectroscopy. We show applications of sech/tanh and WURST pulses to dipolar spectroscopy, including new pulse schemes and procedures, and discuss the more general concept of optimum-control-based pulses for applications in EPR spectroscopy. The article also describes a procedure to correct for experimental imperfections, mostly introduced by the microwave resonator, and discusses further potential applications and limitations of such pulses.

Broadband excitation pulses with variable RF amplitude-dependent flip angle (RADFA)

Pulse sequences in NMR spectroscopy sometimes require the adjustment of effective flip angles with respect to experiment-specific or sample-specific parameters. Here, we present a quality factor for efficient optimization of offset-compensated broadband excitation pulses with RF amplitude-dependent effective flip angles (RADFA). After proof of principle, physical limits of RF amplitude-restricted and RF power-restricted broadband RADFA pulses are explored and corresponding pulse shapes and performances characterized in detail.

Robust INEPT and refocused INEPT transfer with compensation of a wide range of couplings, offsets, and B1-field inhomogeneities (COB3)

Following the two-step optimization procedure previously introduced with the COB-INEPT (Ehni and Luy, 2012), a corresponding inphase-to-antiphase transfer element with close to optimal transfer efficiencies over a coupling range comprising approximately J-6J has been derived. The hard pulse sequence length is only 5.5 ms for coupling constants within 125-750 Hz. Robustness with respect to an offset range of 37.5 kHz on carbon (corresponding to 250 ppm on a 600 MHz spectrometer) and 10 kHz on protons (16.6 ppm at 600 MHz) is achieved with corresponding BUBI and BURBOP broadband pulses. As the sequence achieves a three times higher upper limit of J-compensation compared to the COB-INEPT, we name the transfer element COB3-INEPT. Next to the description of optimization and pulse sequence details, the performance of the resulting element is demonstrated on a test sample and partially aligned sample with actual total couplings in the range of 134 Hz⩽(1)TCH⩽391 Hz. The sequence can also be used for inphase-to-antiphase transfer starting from carbon, where the upper limit of J-compensation is 6J for CH-groups, 3J for CH2-groups, and slightly less than 2J for CH3. Theoretical transfers and experimental verification for the different multiplicities in an refocused INEPT are given.

BEBE(tr) and BUBI: J-compensated concurrent shaped pulses for 1H-13C experiments

Shaped pulses designed for broadband excitation, inversion and refocusing are important tools in modern NMR spectroscopy to achieve robust pulse sequences especially in heteronuclear correlation experiments. A large variety of mostly computer-optimized pulse shapes exist for different desired bandwidths, available rf-field strengths, and tolerance to B1-inhomogeneity. They are usually derived for a single spin 1/2, neglecting evolution due to J-couplings. While pulses with constant resulting phase are selfcompensated for heteronuclear coupling evolution as long as they are applied exclusively on a single nucleus, the situation changes for concurrently applied pulse shapes. Using the example of a (1)H,(13)C two spin system, two J-compensated pulse pairs for the application in INEPT-type transfer elements were optimized: a point-to-point pulse sandwich called BEBE(tr), consisting of a broadband excitation and time-reversed excitation pulse, and a combined universal rotation and point-to-point pulse pair called BUBI, which acts as a refocusing pulse on (1)H and a corresponding inversion pulse on (13)C. After a derivation of quality factors and optimization protocols, a theoretical and experimental comparison with conventionally derived BEBOP, BIBOP, and BURBOP-180° pulses is given. While the overall transfer efficiency of a single pulse pair is only reduced by approximately 0.1%, resulting transfer to undesired coherences is reduced by several percent. In experiments this can lead to undesired phase distortions for pairs of uncompensated pulse shapes and even differences in signal intensities of 5-10% in HSQC and up to 68% in more complex COB-HSQC experiments.

Complete description of the interactions of a quadrupolar nucleus with a radiofrequency field. Implications for data fitting

We present a theory, with experimental tests, that treats exactly the effect of radiofrequency (RF) fields on quadrupolar nuclei, yet retains the symbolic expressions as much as possible. This provides a mathematical model of these interactions that can be easily connected to state-of-the-art optimization methods, so that chemically-important parameters can be extracted from fits to experimental data. Nuclei with spins >1/2 typically experience a Zeeman interaction with the (possibly anisotropic) local static field, a quadrupole interaction and are manipulated with RF fields. Since RF fields are limited by hardware, they seldom dominate the other interactions of these nuclei and so the spectra show unusual dependence on the pulse width used. The theory is tested with (23)Na NMR nutation spectra of a single crystal of sodium nitrate, in which the RF is comparable with the quadrupole coupling and is not necessarily on resonance with any of the transitions. Both the intensity and phase of all three transitions are followed as a function of flip angle. This provides a more rigorous trial than a powder sample where many of the details are averaged out. The formalism is based on a symbolic approach which encompasses all the published results, yet is easily implemented numerically, since no explicit spin operators or their commutators are needed. The classic perturbation results are also easily derived. There are no restrictions or assumptions on the spin of the nucleus or the relative sizes of the interactions, so the results are completely general, going beyond the standard first-order treatments in the literature.

The Fantastic Four: A plug 'n' play set of optimal control pulses for enhancing NMR spectroscopy

We present highly robust, optimal control-based shaped pulses designed to replace all 90° and 180° hard pulses in a given pulse sequence for improved performance. Special attention was devoted to ensuring that the pulses can be simply substituted in a one-to-one fashion for the original hard pulses without any additional modification of the existing sequence. The set of four pulses for each nucleus therefore consists of 90° and 180° point-to-point (PP) and universal rotation (UR) pulses of identical duration. These 1ms pulses provide uniform performance over resonance offsets of 20kHz ((1)H) and 35kHz ((13)C) and tolerate reasonably large radio frequency (RF) inhomogeneity/miscalibration of ±15% ((1)H) and ±10% ((13)C), making them especially suitable for NMR of small-to-medium-sized molecules (for which relaxation effects during the pulse are negligible) at an accessible and widely utilized spectrometer field strength of 600MHz. The experimental performance of conventional hard-pulse sequences is shown to be greatly improved by incorporating the new pulses, each set referred to as the Fantastic Four (Fanta4).

A systematic approach for optimizing the robustness of pulse sequence elements with respect to couplings, offsets, and B1-field inhomogeneities (COB)

Robust experiments that cover a wide range of chemical shift offsets and J-couplings are highly desirable for a multitude of applications in small molecule NMR spectroscopy. Many attempts to improve individual aspects of the robustness of pulse sequence elements based on rational and numerical design have been reported, but a general optimization strategy to cover all necessary aspects for a fully robust sequence is still lacking. In this article, a viable optimization strategy is introduced that covers a defined range of couplings, offsets, and B(1)-field inhomogeneities (COB) in a time-optimal way. Individual components of the optimization strategy can be optimized in any adequate way. As an example for the COB approach, we present the (1)H -(13)C-COB-INEPT with transfer of approximately 99% over the full carbon and proton bandwidth and (1)J(CH) -couplings in the range of 120-250 Hz, which have been optimized using efficient algorithms derived from optimal control theory. The theoretical performance is demonstrated in a number of corresponding COB-HSQC experiments.

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

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 B(1)-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.

Slice-selective broadband refocusing pulses for the robust generation of crushed spin-echoes

A major challenge for in vivo magnetic resonance spectroscopy with point-resolved spectroscopy (PRESS) is the low signal intensity for the measurement of weakly scalar coupled spins, for example lactate. The chemical-shift displacement error between the two coupling partners of the lactate molecule leads to a signal decrease. The chemical-shift displacement error is decreased and therefore the lactate signal is increased by using refocusing pulses with a broad bandwidth. Previously, slice-selective broadband universal rotation pulses (S-BURBOP) were designed and applied as refocusing pulses in the PRESS pulse sequence (Janich MA, et al., Journal of Magnetic Resonance, 2011, 213, 126-135). However, S-BURBOP pulses leave a phase error across the slice which is superimposed on the spectra when spatially resolving the PRESS voxel. In the present novel design of slice-selective broadband refocusing pulses (S-BREBOP) this phase error is avoided. S-BREBOP pulses obtain 2.5 times the bandwidth of conventional Shinnar-Le Roux pulses and are robust against ±20% miscalibration of the B(1) amplitude. S-BREBOP pulses were validated in phantoms and in a low-grade brain tumor of a patient. Compared to conventional Shinnar-Le Roux pulses they lead to a decrease of the chemical-shift displacement error and consequently a lactate signal increase.

Tailored real-time scaling of heteronuclear couplings

Heteronuclear couplings are a valuable source of molecular information, which is measured from the multiplet splittings of an NMR spectrum. Radiofrequency irradiation on one coupled nuclear spin allows to modify the effective coupling constant, scaling down the multiplet splittings in the spectrum observed at the resonance frequency of the other nuclear spin. Such decoupling sequences are often used to collapse a multiplet into a singlet and can therefore simplify NMR spectra significantly. Continuous-wave (cw) decoupling has an intrinsic non-linear offset dependence of the scaling of the effective J-coupling constant. Using optimal control pulse optimization, we show that virtually arbitrary off-resonance scaling of the J-coupling constant can be achieved. The new class of tailored decoupling pulses is named SHOT (Scaling of Heteronuclear couplings by Optimal Tracking). Complementing cw irradiation, SHOT pulses offer an alternative approach of encoding chemical shift information indirectly through off-resonance decoupling, which however makes it possible for the first time to achieve linear J scaling as a function of offset frequency. For a simple mixture of eight aromatic compounds, it is demonstrated experimentally that a 1D-SHOT {(1)H}-(13)C experiment yields comparable information to a 2D-HSQC and can give full assignment of all coupled spins.

Designing optimal universal pulses using second-order, large-scale, non-linear optimization

Recently, RF pulse design using first-order and quasi-second-order pulses has been actively investigated. We present a full second-order design method capable of incorporating relaxation, inhomogeneity in B(0) and B(1). Our model is formulated as a generic optimization problem making it easy to incorporate diverse pulse sequence features. To tame the computational cost, we present a method of calculating second derivatives in at most a constant multiple of the first derivative calculation time, this is further accelerated by using symbolic solutions of the Bloch equations. We illustrate the relative merits and performance of quasi-Newton and full second-order optimization with a series of examples, showing that even a pulse already optimized using other methods can be visibly improved. To be useful in CPMG experiments, a universal refocusing pulse should be independent of the delay time and insensitive of the relaxation time and RF inhomogeneity. We design such a pulse and show that, using it, we can obtain reliable R(2) measurements for offsets within ±γB(1). Finally, we compare our optimal refocusing pulse with other published refocusing pulses by doing CPMG experiments.

Shaped optimal control pulses for increased excitation bandwidth in EPR

A 1 ns resolution pulse shaping unit has been developed for pulsed EPR spectroscopy to enable 14-bit amplitude and phase modulation. Shaped broadband excitation pulses designed using optimal control theory (OCT) have been tested with this device at X-band frequency (9 GHz). FT-EPR experiments on organic radicals in solution have been performed with the new pulses, designed for uniform excitation over a significantly increased bandwidth compared to a classical rectangular π/2 pulse of the same B(1) amplitude. The concept of a dead-time compensated prefocused pulse has been introduced to EPR with a self-refocusing of 200 ns after the end of the pulse. Echo-like refocused signals have been recorded and compared to the performance of a classical Hahn-echo sequence. The impulse response function of the microwave setup has been measured and incorporated into the algorithm for designing OCT pulses, resulting in further significant improvements in performance. Experimental limitations and potential new applications of OCT pulses in EPR spectroscopy will be discussed.

Optimal control design of band-selective excitation pulses that accommodate relaxation and RF inhomogeneity

Existing optimal control protocols for mitigating the effects of relaxation and/or RF inhomogeneity on broadband pulse performance are extended to the more difficult problem of designing robust, refocused, frequency selective excitation pulses. For the demanding case of T(1) and T(2) equal to the pulse length, anticipated signal losses can be significantly reduced while achieving nearly ideal frequency selectivity. Improvements in performance are the result of allowing residual unrefocused magnetization after applying relaxation-compensated selective excitation by optimized pulses (RC-SEBOPs). We demonstrate simple pulse sequence elements for eliminating this unwanted residual signal.

New strategies for designing robust universal rotation pulses: application to broadband refocusing at low power

Optimizing pulse performance often requires a compromise between maximizing signal amplitude and minimizing spectral phase errors. We consider methods for the de novo design of universal rotation pulses, applied specifically but not limited to refocusing pulses. 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 methodologies for generating pulses suitable for broadband refocusing are considered. The de novo design of 180° universal rotation pulses (180(UR)(°)) using optimal control can provide improved performance compared to schemes which construct refocusing pulses as composites of existing pulses. The advantages of broadband universal rotation by optimized pulses (BURBOP) are most evident for pulse design that includes tolerance to RF inhomogeneity or miscalibration. Nearly ideal refocusing is possible over a resonance offset range of ± 170% relative to the nominal pulse B(1) field, concurrent with tolerance to B(1) inhomogeneity/miscalibration of ± 33%. We present new modifications of the optimal control algorithm that incorporate symmetry principles (S-BURBOP) and relax conservative limits on peak RF pulse amplitude for short time periods that pose no threat to the probe. We apply them to generate a set of low-power 180(BURBOP)(°) pulses suitable for widespread use in (13)C spectroscopy on the majority of available probes. A quantitative measure for the reduced spectral phase error provided by these symmetry principles is also derived. For pulses designed according to this symmetry, refocusing phase errors are virtually eliminated upon application of EXORCYCLE or an equivalent G-180(S-BURBOP)(°)-G gradient sandwich, independent of resonance offset and RF inhomogeneity. The magnitude of the refocused component is not significantly compromised in achieving such ideal phase performance.

Design and application of robust rf pulses for toroid cavity NMR spectroscopy

We present robust radio frequency (rf) pulses that tolerate a factor of six inhomogeneity in the B₁ field, significantly enhancing the potential of toroid cavity resonators for NMR spectroscopic applications. Both point-to-point (PP) and unitary rotation (UR) pulses were optimized for excitation, inversion, and refocusing using the gradient ascent pulse engineering (GRAPE) algorithm based on optimal control theory. In addition, the optimized parameterization (OP) algorithm applied to the adiabatic BIR-4 UR pulse scheme enabled ultra-short (50 μs) pulses with acceptable performance compared to standard implementations. OP also discovered a new class of non-adiabatic pulse shapes with improved performance within the BIR-4 framework. However, none of the OP-BIR4 pulses are competitive with the more generally optimized UR pulses. The advantages of the new pulses are demonstrated in simulations and experiments. In particular, the DQF COSY result presented here represents the first implementation of 2D NMR spectroscopy using a toroid probe.

Cooperative pulses

We introduce the concept of cooperative (COOP) pulses which are designed to compensate each other's imperfections. In multi-scan experiments, COOP pulses can cancel undesired signal contributions, complementing and generalizing phase cycles. COOP pulses can be efficiently optimized using an extended version of the optimal-control-based gradient ascent pulse engineering (GRAPE) algorithm. The advantage of the COOP approach is experimentally demonstrated for broadband and band-selective pulses.

Optimal control design of pulse shapes as analytic functions

Representing NMR pulse shapes by analytic functions is widely employed in procedures for optimizing performance. Insights concerning pulse dynamics can be applied to the choice of appropriate functions that target specific performance criteria, focusing the solution search and reducing the space of possible pulse shapes that must be considered to a manageable level. Optimal control theory can accommodate significantly larger parameter spaces and has been able to tackle problems of much larger scope than more traditional optimization methods. However, its numerically generated pulses, as currently constructed, do not readily incorporate the capabilities of particular functional forms, and the pulses are not guaranteed to vary smoothly in time, which can be a problem for faithful implementation on older hardware. An optimal control methodology is derived for generating pulse shapes as simple parameterized functions. It combines the benefits of analytic and numerical protocols in a single powerful algorithm that both complements and enhances existing optimization strategies.

Heteronuclear decoupling by optimal tracking

The problem to design efficient heteronuclear decoupling sequences is studied using optimal control methods. A generalized version of the gradient ascent engineering (GRAPE) algorithm is presented that makes it possible to design complex non-periodic decoupling sequences which are characterized by tens of thousands of pulse sequence parameters. In contrast to conventional approaches based on average Hamiltonian theory, the concept of optimal tracking is used: a pulse sequence is designed that steers the evolution of an ensemble of spin systems such that at a series of time points, a specified trajectory of the density operator is tracked as closely as possible. The approach is demonstrated for the case of low-power heteronuclear decoupling in the liquid state for in vivo applications. Compared to conventional sequences, significant gains in decoupling efficiency and robustness with respect to offset and inhomogeneity of the radio-frequency field were found in simulations and experiments.

Exploring the limits of broadband excitation and inversion: II. Rf-power optimized pulses

In [K. Kobzar, T.E. Skinner, N. Khaneja, S.J. Glaser, B. Luy, Exploring the limits of broadband excitation and inversion, J. Magn. Reson. 170 (2004) 236-243], optimal control theory was employed in a systematic study to establish physical limits for the minimum rf-amplitudes required in broadband excitation and inversion pulses. In a number of cases, however, experimental schemes are not limited by rf-amplitudes, but by the overall rf-power applied to a sample. We therefore conducted a second systematic study of excitation and inversion pulses of varying pulse durations with respect to bandwidth and rf-tolerances, but this time using a modified algorithm involving restricted rf-power. The resulting pulses display a variety of pulse shapes with highly modulated rf-amplitudes and generally show better performance than corresponding pulses with identical pulse length and rf-power, but limited rf-amplitude. A detailed description of pulse shapes and their performance is given for the so-called power-BEBOP and power-BIBOP pulses.

Linear phase slope in pulse design: application to coherence transfer

Using optimal control methods, robust broadband excitation pulses can be designed with a defined linear phase dispersion. Applications include increased bandwidth for a given pulse length compared to equivalent pulses requiring no phase correction, selective pulses, and pulses that mitigate the effects of relaxation. This also makes it possible to create pulses that are equivalent to ideal hard pulses followed by an effective evolution period. For example, in applications, where the excitation pulse is followed by a constant delay, e.g. for the evolution of heteronuclear couplings, part of the pulse duration can be absorbed in existing delays, significantly reducing the time overhead of long, highly robust pulses. We refer to the class of such excitation pulses with a defined linear phase dispersion as ICEBERG pulses (Inherent Coherence Evolution optimized Broadband Excitation Resulting in constant phase Gradients). A systematic study of the dependence of the excitation efficiency on the phase dispersion of the excitation pulses is presented, which reveals surprising opportunities for improved pulse sequence performance.

Optimal control design of excitation pulses that accommodate relaxation

An optimal control algorithm for mitigating the effects of T(1) and T(2) relaxation during the application of long pulses is derived. The methodology is applied to obtain broadband excitation that is not only tolerant to RF inhomogeneity typical of high resolution probes, but is relatively insensitive to relaxation effects for T(1) and T(2) equal to the pulse length. The utility of designing pulses to produce specific phase in the final magnetization is also presented. The results regarding relaxation and optimized phase are quite general, with many potential applications beyond the specific examples presented here.

P.E.HSQC: a simple experiment for simultaneous and sign-sensitive measurement of (1JCH+DCH) and (2JHH+DHH) couplings

The angular information content of residual dipolar couplings between nuclei of fixed distance makes the accurate and sign-sensitive measurement of (1JCH+DCH) and (2JHH+DHH) couplings highly desirable. Experiments published so far are typically highly specialized for the effective measurement of a subset of couplings. The P.E.HSQC presented here, is an E.COSY based experiment which allows the simultaneous measurement of all heteronuclear and homonuclear couplings within CH, CH2, and CH3 groups in a single spectrum with the necessary precision and sign information. The simplicity of the approach and the absence of artefacts like phase distortions due to antiphase evolution make it ideally suited for coupling determination of organic molecules at natural abundance.

Exploring the limits of polarization transfer efficiency in homonuclear three spin systems

The limits of polarization transfer efficiency are explored for systems consisting of three isotropically coupled spins 1/2 in the absence of relaxation. An idealized free evolution and control Hamiltonian is studied, which provides an upper limit of transfer efficiency (in terms of transfer amplitude and transfer time) for realistic homonuclear spin systems with arbitrary Heisenberg-type coupling constants J12, J13, and J23. It is shown that optimal control based pulse sequences have significantly improved transfer efficiencies compared to conventional transfer schemes. An experimental demonstration of optimal polarization transfer is given for the case of the carbon spin system of fully 13C labelled alanine at 62.5 MHz Larmor frequency.

Optimal control design of constant amplitude phase-modulated pulses: application to calibration-free broadband excitation

An optimal control algorithm for generating purely phase-modulated pulses is derived. The methodology is applied to obtain broadband excitation with unprecedented tolerance to RF inhomogeneity. Design criteria were transformation of Iz-->Ix over resonance offsets of +/-25 kHz for constant RF amplitude anywhere in the range 10-20 kHz, with a pulse length of 1 ms. Simulations transform Iz to greater than 0.99 Ix over the targetted ranges of resonance offset and RF variability. Phase deviations in the final magnetization are less than 2-3 degrees over almost the entire range, with sporadic deviations of 6-9 degrees at a few offsets for the lowest RF (10 kHz) in the optimized range. Experimental performance of the new pulse is in excellent agreement with the simulations, and the robustness of the excitation pulse and a derived refocusing pulse are demonstrated by insertion into conventional HSQC and HMBC-type experiments.

Construction of universal rotations from point-to-point transformations

For a desired range of offsets, universal rotations of arbitrary flip angle can be constructed based on point-to-point rotations of I(y) with half the flip angle. This approach allows, for example, creation of broadband or bandselective refocusing pulses from broadband or bandselective excitation pulses. Furthermore, universal rotations about any axis can be obtained from point-to-point transformations that can easily be optimized using optimal control algorithms. The construction procedure is demonstrated on the examples of a broadband refocusing pulse, a broadband 120(x) degrees rotation and a z-rotation with offset pattern.

Pattern pulses: design of arbitrary excitation profiles as a function of pulse amplitude and offset

A novel class of pulses is presented which can be regarded as a generalization of both frequency-selective pulses and B1-selective pulses. The excitation profile of these pulses forms a pre-defined pattern in two dimensions, which are spanned by pulse offset and radio-frequency (RF) amplitude. The presented pulses were designed numerically based on principles of optimal control theory. For simple test patterns, we demonstrate the flexibility of this approach by simulations and experiments. This previously unknown flexibility may trigger novel applications in NMR spectroscopy and imaging. As a first practical application, we demonstrate a direct approach for calibrating RF pulses.