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

Abstract 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 19F -PROJECT experiment for measuring relaxation times with reduced distortions due to J -coupling evolution.


Classification of Shaped Pulses
. Effective rotations are illustrated as normalized vectors for various offsets.
Shaped pulses are composed of numerous much shorter pulses that can be modulated in phase and amplitude.The effective propagator   of a shaped pulse, consisting of  steps, can hence be described by a piece-wise time-independent propagation: Such a propagator   represents an "effective rotation" that can be used for pulse shape classification as illustrated in Figure S1  For a "matched" pulse pair of a SORDOR-180 (B) and a SORDOR-90 (B1) the effective rotations are distributed identically in the transverse plane or in other words   is the same for the SORDOR-180 and the SORDOR-90.For this reason the pulse pair can be used as universal rotations with quadratic phase distribution.
Exponential Fits: Figure 5 Signals of Figure 5 were extracted and fitted to an exponential decay using python.Due to strong coupling, CF2 groups exhibit phase distortions and exponential fits show large residuals (red box).

Exponential Fits: Comparison to On-Resonant Hard Pulse
Exponential decays of the PROJECT experiment using SORDOR pulse pairs are compared to onresonant hard pulses for the Bruker "doped water" standard sample (0.1mg/ml GdCl3, 1% H2O and 0.1% 13 C-labeled CH3OH in D2O).Since the water signal is not subject to coupling evolution the decay rate does not depend on the mixing for J refocusing induced by PROJECT and it is possible to estimate that SORDOR pulse imperfections have only a minor influence on the relaxation measurement.SORDOR pulses are also used on-resonant, but a much larger bandwidth is expected (as shown in Figure 5 of main text).

Pulse program for SORDOR Echo
In order to execute python scripts in Topspin, please run the "edpy" command in Topspin and create new python scripts called pk2.py (1D version) and pk22D.py(pseudo-2D version).Copy and paste the subsequent code to the created scripts and execute from the Topspin command line.For a 1D, please run "pk2 " (e.g."pk2 3.65"), where  is the value for second order phase correction and the phase  is calculated from where  is the spectral width and  is the frequency offset.Setting the spectral width to the bandwidth covered by SORDOR pulses can facilitate the procedure.For a pseudo-2D (e.g. 19F-PROJECT), the second order phase correction is applied successively and the pseudo-2D is stored as multiple 1Ds.The script is executed with "pk22D  procno", where procno is the processing number of the first 1D spectrum and following 1Ds are stored in ascending numbers.
and S2 for different 180° and 90° pulses, respectively.The most versatile class A of 180° pulses is shown in Fig S1 (a) where a rotation about the x-axis is achieved for all considered offsets.Shaped 180° pulses of class A are typically referred to as "refocusing pulse" or "universal rotations" (UR) with   =  180 .Shaped 180° pulses of class B, on the other hand, induce phase shifts for transverse components but can be used as an inversion of the zcomponent -the effective rotations are shown in Fig S1 (b).These shaped 180° pulses are generally referred to as "point-to-point" or "inversion pulses" and   =    180   † where   corresponds to an arbitrary z-rotation.For shaped 90° pulses a more distinctive classification is required as shown in Fig S2.Shaped 90° pulses of class A induce a 90° rotation about a single axis for all considered offsets and are, hence, also referred to as "universal rotations" where   =  90 in Fig S2 (a).Effective rotations for shaped 90° pulses of class B1 are illustrated in Fig S2 (b) where offset-dependent rotation axes are distributed in the transverse plane with   =    90   † .With respect to SORDOR-90 pulses, it is crucial to note that this class of 90° pulses can be considered universal rotations where an offset dependent phase is acquired.Pulse shapes of class B2, on the other hand, can only be used to transfer a single component of magnetization.Therefore, B2-pulses are typically used to excite z-magnetization to the transverse plane with defined phase (e.g. to  ̂) as shown in Fig S2 (c) with    ̂  † =  ̂.It is noteworthy, that for these so-called "point-to-point" or "excitation pulses" the effective rotations are in a tilted plane.This is based on the fact that a transfer from  ̂ to  ̂ can be obtained e.g. by a 90° rotation about x but also by a 270° degree rotation about -x and further by a 180° rotation about the axes in the intersection of the tilted and the yz-plane.An excitation of z-magnetization to a state with undefined phase can be obtained from shaped pulses of class B3 where    ̂  † =    ̂  † and   corresponds to an arbitrary z-rotation.In Fig S2 (d) certain effective rotation axes are shown but in principle all axes in between the two cones are thinkable.