Perspectives of shaped pulses for EPR spectroscopy

Journal of Magnetic Resonance 280 (2017) 30–45

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Perspectives in Magnetic Resonance

Perspectives of shaped pulses for EPR spectroscopy Philipp E. Spindler a, Philipp Schöps a, Wolfgang Kallies b, Steffen J. Glaser b,⇑, Thomas F. Prisner a,⇑ a b

Institute of Physical and Theoretical Chemistry and Center of Biomolecular Magnetic Resonance, Goethe University Frankfurt, Germany Department of Chemistry, Technical University of Munich, Germany

a r t i c l e

i n f o

Article history: Received 18 December 2016 Revised 27 February 2017 Accepted 28 February 2017

Keywords: Shaped pulses Adiabatic pulses OCT pulses Optimum control pulses Dipolar EPR spectroscopy

a b s t r a c t 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-controlbased 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. Ó 2017 Elsevier Inc. All rights reserved.

1. Introduction ‘‘Nothing tends so much to the advancement of knowledge as the application of a new instrument. The native intellectual powers of men in different times are not so much the causes of different success of their labours, as to the particular nature of the means of artificial resources in their possession” [1]. This citation from Sir Humphry Davy dating back more than 200 years describes exquisitely the advances that have occurred in the field of Electron Paramagnetic Resonance (EPR) spectroscopy. Many of these advances were driven (or hampered) by technological advances (or limitations). This holds true for fast data acquisition and averaging, for high frequency EPR in the THz regime as well as for the possibility of ultrafast amplitude and phase manipulation of microwave pulses, which is the subject of this perspective article. As electron spins possess magnetic moments three orders of magnitude larger than those of nuclear spins, the technical requirements for manipulating electron spins in EPR are much more demanding, compared to the manipulation of nuclear spins in the field of Nuclear Magnetic Resonance (NMR) spectroscopy, where techniques using phase and amplitude modulated pulses have been in use since the 1980s [2–8]. Manipulation of nuclear spin systems usually requires excitation bandwidths in the hundred kilohertz regime, whereas the spectral width of paramagnetic samples is typically in the hundred megahertz regime for organic radicals and might exceed gigahertz for transition metals in a magnetic ⇑ Corresponding authors. E-mail address: [email protected] (T.F. Prisner). http://dx.doi.org/10.1016/j.jmr.2017.02.023 1090-7807/Ó 2017 Elsevier Inc. All rights reserved.

field of only 1 T. The spectral widths become even larger at higher magnetic field strengths due to g-tensor anisotropy. Furthermore, the transverse relaxation times of electron spins are typically in the low microsecond regime. Therefore, amplitude and phase modulation of pulses for EPR spectroscopy has to be performed on the nanosecond time scale to be efficient. It is only recently that versatile arbitrary waveform generators (AWGs) with clock rates higher than a gigahertz have become readily available, with this newfound availability driven mostly by the tremendous advances in semiconductor technology for digital communication. As the necessary technology for coherent phase- and amplitude manipulation of electron spins on a gigahertz rate is now available, many of the concepts developed in the field of NMR can be adapted and explored in the field of EPR spectroscopy. These include pseudo-stochastic, composite, adiabatic, frequency chirped or amplitude- and phase-modulated shaped pulses, as well as the methodologies to design pulse sequences for specific purposes, for example optimum control theory (OCT) [9–25]. In most cases a direct ‘‘copy and paste” of NMR pulse sequences is not appropriate due to the different structure of the spinHamiltonians found in the fields of NMR and EPR. In liquid-state NMR the spin-Hamiltonian only consists of isotropic couplings between nuclear spins, whereas in the fields of EPR and solidstate NMR a network of strongly dipolar coupled spins exists. Two important areas for applications in the field of pulsed EPR are hyperfine and dipolar spectroscopies. In hyperfine spectroscopy the pairwise interactions of an isolated unpaired electron spin with nuclear spins in its close vicinity (<1 nm) are explored. Pulsed dipolar spectroscopy (PDS) is used to measure the coupling

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between two paramagnetic spin labels separated in the 1.5–10 nm range [26,27]. Both types of spectroscopy use experiments performed on solid state samples and therefore they are subject to anisotropic interactions (g- and hyperfine tensors, dipolar and zero-field splitting). Additionally, high-spin paramagnetic centers, for example Gd3+ (S = 7/2) or Mn2+ (S = 5/2), can further complicate the spin-Hamiltonian. More complications result if more than two spins are dipolar coupled, as often is the case for oligomeric protein complexes. EPR also differs compared to NMR spectroscopy in its technical requirements and restrictions. As stated previously, the phase and amplitude manipulations of pulses for use in EPR have to be performed on a much shorter timescale compared with NMR. At the same time, the intrinsic microwave resonator bandwidth limits the switching time and efficiency of the pulses. The resonator bandwidth can only be increased by sacrificing detection sensitivity; at least at typically used microwave frequencies (X-band or Qband, 9.5 and 34 GHz respectively). Therefore, a compromise between excitation efficiency and detection sensitivity has to be achieved. In this perspective we will focus on some recent applications of our two research groups in the field of dipolar EPR spectroscopy with shaped pulses, discuss advantages and technical limitations of such pulses and the potential of optimum control theory for the optimization of EPR pulse sequences. First, we will briefly describe the characteristics of two classes of monotonously frequency-swept pulses, which have been used by us for pulsed dipolar spectroscopy experiments. Next, we will include descriptions of the experimental applications of these pulses. Adiabatic inversion of EPR spins will be initially considered, followed by the use of such pulses to create transversal magnetization. After that, the potential of more complicated pulses, derived by OCT, will be discussed and illustrated by experimental EPR results. We include a short discussion on the control and optimization of the experimental performance of such pulses and conclude with a discussion of the potential of future applications. 2. Description of adiabatic and non-adiabatic broadband pulses Adiabatic pulses have a p flip angle over a well-defined bandwidth independent of the B1 microwave magnetic field strength, provided that B1 surpasses a certain threshold value. Hence, adiabatic pulses can be used to compensate for B1 inhomogeneity caused by hardware limitations [28]. As described in [29], this compensation property is used to generate high pulse bandwidths and uniform inversion profiles. Furthermore, the manipulation of spins outside the well-defined bandwidth is negligible (see Fig. 1). These properties have made adiabatic pulses very attractive for an increasing number of applications in EPR spectroscopy. The microwave magnetic field component of an arbitrary shaped pulse in the rotating frame can be defined as:

0

cosðuðtÞ þ u0 Þ

1

B C B1 ðtÞ ¼ B1 ðtÞ@ sinðuðtÞ þ u0 Þ A

ð1Þ

0 with B1(t) being the amplitude, u(t) the phase function and u0 the constant phase factor. For an adiabatic pulse, the phase function is usually given as:

uðtÞ ¼

Z

t

t p =2

DxMW ðt 0 Þdt0

ð2Þ

with tp being the pulse duration and DxMW(t) the instantaneous microwave frequency offset, which is defined as the difference between the instantaneous microwave frequency xMW ðtÞ and the center frequency xc of the pulse: DxMW(t) = xMW(t)  xc. In the lit-

erature, usually a combination of Eqs. (1) and (2) is used to define a certain adiabatic pulse, as the amplitude and instantaneous microwave frequency offset are given. Two classes of adiabatic pulses have been used by us, namely the WURST (wideband, uniform rate, smooth truncation) [28] and sech/tanh pulse [30] and a combination of the two types. The WURST pulse is given as follows:

DxMW ðtÞ ¼ kt   n    pt  B1 ðtÞ ¼ B1max 1  sin tp 

ð3Þ

with t 2 [tp/2, tp/2], k being the sweep parameter, B1max the maximum amplitude and n an adjustable parameter, which is used to round off the edges of the inversion profile. An example of a WURST pulse and the corresponding inversion profile can be seen in Fig. 1 (b) and (c). The sech/tanh pulses belong to the second adiabatic pulse family used:

DxMW ðtÞ ¼ 0:5  BWtanhðbtÞ B1 ðtÞ ¼ B1max sechðbtÞ

ð4Þ

with BW being the bandwidth in the case of infinite pulse duration and b an adjustable parameter, which is used to truncate the microwave pulse in the time domain. An example of a sech/tanh pulse and the corresponding inversion profile is shown in Fig. 1 (a) and (c). As shown in Fig. 1(c), the bandwidth of a WURST pulse is larger for a given pulse length and microwave power, whereas the selectivity of the sech/tanh exceeds that of other adiabatic pulses. In order to generate transverse magnetization the pulse shapes defined by Eqs. (3) and (4) can be used. However, either the pulse duration needs to be shortened (t 2 [0, tp/2]), or the maximum microwave power needs to be reduced. Pulses formed using shorter pulse durations are called adiabatic half-passage (AHP) and pulse shapes derived with reduced microwave power are known as fast-passage pulses. The reduced microwave amplitude for a linear swept pulse in dependency of the flip angle h < p can be calculated as follows [31]:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi 2k

ce B1max ðhÞ 1 cosðhÞ þ 1 ¼ ln 2p 2p 2

p

ð5Þ

An example of a p/2 pulse derived with Eq. (5) can be seen in Fig. 1(d). A more detailed description of adiabatic and nonadiabatic broadband pulses in EPR spectroscopy can be found in [29]. 3. Selected applications in dipolar EPR spectroscopy 3.1. Broadband PELDOR with adiabatic inversion pulses PELDOR (Pulsed Electron Electron Double Resonance) [32,33] or DEER (Double Electron Electron Resonance) [34] is an experiment in which the signal-to-noise ratio (S/N) is directly correlated to the bandwidth of the used inversion pump pulse. A Hahn or refocused echo is observed while dipolar coupled spins with resonance frequencies outside the observer frequency window are inverted by a pump pulse on a second frequency. The echo signal is modulated as a function of the pump pulse temporal position within the pulse sequence by the dipolar coupling frequency between the two spins. The depth of the dipolar modulation, and therefore the S/N, is proportional to the fraction of coupled spins inverted by the pump pulse. An easy way to increase the bandwidth of microwave pulses by means of modulation techniques is a frequency sweep during the pulse. As the frequency of the pulse is swept, different spins are subsequently on resonance with the microwave field and can therefore be efficiently manipulated over a much larger frequency

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b

30

20

B1 [2 MHz]

10 0 -10

e

e

B1 [2 MHz]

20

0 -10 -20

-30

-30 50

100

150

200

0

Time [ns]

d

1

100

150

200

0

1

Time [ns]

50

0 -0.5 -1 -200

50

Time [ns]

0.5

M z/M 0

10

-20

0

c

30

-100

0

100

200

Frequency [MHz]

0.5

M z/M 0

a

100

0

150

-0.5

200 -200

-100

0

100

200

Frequency [MHz]

Fig. 1. Comparison of a WURST, a sech/tanh and a rectangular pulse. The time domain shapes of the WURST and sech/tanh pulse in the rotating frame are shown in (a) and (b), respectively. Here x and y components are shown as red and blue lines and the corresponding amplitude functions as a green line for the WURST pulse in (a) and a black line for the sech/tanh pulse in (b). (c) Compares the frequency domain inversion profiles of a rectangular p (dotted blue line), WURST p (green line), WURST p/2 (magenta line) and sech/tanh p pulse (black line). The pulse length used are 16 ns for the rectangular p pulse and 200 ns for the WURST and sech/tanh pulses. For all the p pulses the same maximum microwave field strength of ceB1max/2p = 31.25 MHz was used and for the WURST p/2 pulse a field of ceB1max/2p = 10.78 MHz. (d) Details the Mz component as a function of frequency offset and time for the p WURST pulse. The parameters used to define the pulses were: for the adiabatic WURST pulse n = 10, k = 0.009 GHz/ns for the p and k = 0.0104 GHz/ns for the p/2 pulse, whereas the sech/tanh pulse was calculated with b = 0.043 ns1 and 0.5BW = 0.3142 rad ns1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

range than possible by monochromatic rectangular pulses. Adiabatic pulses show a unique behavior among frequency-swept pulses in that the inversion efficiency is independent of the microwave field strength B1 (above a threshold value) and hence are ideally suited for population inversion [28]. Additionally, although pulse schemes like BIR 1 or BIR 4 [35], not to be confused with the fast passage pulses mentioned previously, can be used to excite transverse magnetization in an adiabatic manner, their bandwidth is roughly limited to the on-resonance nutation frequency. This is not the case for inversion of the spin system using adiabatic pulses, where the pulse bandwidth can be much larger than the excitation field strength. Both of these features make adiabatic inversion pulses very robust against hardware imperfections and therefore suitable for EPR applications. In our first application of adiabatic pulses, we used a modified sech/tanh pulse as a pump pulse for PELDOR [12]. In order to keep the pulse duration short the pulse was truncated by choosing b ¼ t4p , instead of the more commonly used value of b ¼ 10:6 . PELDOR tp experiments using these pump pulses showed substantial increases in modulation depth compared to rectangular pump pulses using the peak B1 field of the sech/tanh pulse (see Fig. 2(d)). Adiabatic pulses are always longer than rectangular pulses, thus shortening the achievable observation time window for the dipolar evolution since a temporal overlap of pulses has to be avoided. A second disadvantage arises from the frequency modulation of the adiabatic pulses which causes spins with different resonance fre-

quencies to be inverted at different times during the pulse, as shown in Fig. 1(d). This leads to different dipolar evolution times for spins with different resonance frequencies, truncating the dipolar Pake pattern at higher frequencies (see Fig. 2(e)). It is not easy to give a general rule stating the minimum inter-spin distance with which it is suitable to use an adiabatic pulse of a certain length as this is dependent on all the parameters defining the pulse rather than solely the pulse length. Closer inspection of the sech/tanh pulse shows that, actual inversion of the spin packets takes place in a time window much shorter than the total pulse duration, the length of which is dependent on the b parameter. Acceptable conditions are met if the actual duration of inversion is smaller than the period of the perpendicular dipolar oscillation of the Pake pattern corresponding to the inter-spin distance. A common technique used in NMR is to refocus the dispersion caused by one frequency-swept pulse using a second timereversed pulse with an opposite sweep direction. This can also be applied in the field of EPR [12]. Comparison of the PELDOR time traces with one and two inversion pulses on a Co2+-nitroxide model system (Fig. 2(g) and (h)) demonstrates, that indeed the damping of the dipolar oscillation by the prolonged pump pulse length can be suppressed by using a second adiabatic inversion pulse with opposite frequency sweep direction. However, an artifact at the end of the time trace is visible in this case. This artifact is caused by spins having been inverted by only one of the pump pulses. Therefore, the observed signal becomes a superposition of

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Fig. 2. (a) Co-nitroxide model compound used for all PELDOR measurements shown in this figure. (b) Pump and probe spectral positions (600 MHz separation). (c) 4-pulse PELDOR sequence with one broadband inversion pulse. The colors (red and blue) represent the quadrature components of the pulses used for detection (x and y pulse phases) and pumping (IQ modulator drive signals). (d) 4-pulse PELDOR raw data obtained with a 150 MHz bandwidth sech/tanh pulse (blue line) compared to a 20 ns rectangular 4 pump pulse (black line). The sech/tanh pulse was calculated with b ¼ 200 ns1 and t p ¼ 200ns. (e) Fourier transformations of the data shown in (d) after removal of the intermolecular background component, with data recorded using a sech/tanh broadband pump pulse (blue line) and rectangular pump pulse (black line). (f) Broadband 5pulse PELDOR sequence. The color coding of pulses is the same as for (c). (g) Broadband 5-pulse PELDOR raw data (blue line) compared to 4-pulse PELDOR data (black line). Both are recorded with 50 MHz sech/tanh pump pulses. (h) Fourier transformations of the data shown in (g) after removal of the intermolecular background component with broadband 5-pulse PELDOR data (blue line) and broadband 4-pulse PELDOR data (black line). (i) IQ modulator drive signals (red and blue for different quadrature components) of the 400 MHz WURST pulse used as a pump pulse for the broadband 5-pulse PELDOR shown in (k) which also uses a 480 MHz WURST pulse for refocusing. The 400 and 480 MHz WURST pulses both utilize a sweep rate of 1.12 MHz/ns and n = 30; (j) Comparison of inversion profiles of a 20 ns rectangular pulse (black line) with the 400 MHz WURST pulse (blue line) and the 480 MHz WURST pulse (green line). (k) Comparison between broadband 5-pulse PELDOR with a 400 MHz WURST pump pulse and a 480 MHz WURST refocusing pulse (blue line) and 4-pulse PELDOR with a 20 ns rectangular pump pulse (black line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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dipolar evolutions arising from spins affected by either one or two inversion pulses. This artifact signal can be suppressed by increasing the bandwidth of the second pump pulse relative to the bandwidth of the first pump pulse. The refocused echo in the 4-pulse PELDOR sequence has two refocusing periods and one pump pulse, this limits the dipolar evolution time to the longer refocusing period. 5-pulse PELDOR utilizes the same refocused echo sequence but the inclusion of two pump pulses enables the dipolar evolution window to be the sum of both refocusing periods (Fig. 2(f)). The application of two broadband inversion pump pulses hence increases the achievable time window by approximately the inter-pulse delay between the first and second observer pulse, but makes the experimental settings of course somewhat more demanding. Typical experimental setups for PELDOR experiments consist of a 1 kW TWT amplifier at X-band or a 150 W TWT at Qband frequencies. These allow to achieve a modulation depth of k = 0.5–0.6 for rectangular pump pulses acting on nitroxide spin labels. A shaped pump pulse can increase the modulation depth to k = 0.7–0.8. The gain in modulation depth is of course more strongly enhanced when metal centers with broad EPR spectra are involved, as demonstrated with the Co2+-nitroxide biradical in Fig. 2. This can be most impressively seen in Fig. 2(k), where a 5-pulse PELDOR experiment with two WURST pulses (400 and 480 MHz width) is compared with a 4-pulse PELDOR experiment with a rectangular monochromatic inversion pulse. 3.2. CP-PELDOR sequence The maximum distance that can be extracted from PELDOR data depends strongly on the maximum achievable dipolar evolution time window. The time window must be sufficiently long that not only can the full Pake pattern be detected with enough frequency resolution; more importantly, it has to be long enough to be able to separate the intermolecular contribution to the signal from the intramolecular part of interest. The maximum time window of an experiment is restricted by the dephasing time of the transversal magnetization. It has been shown that proton spin diffusion is one of the major relaxation pathways for the transversal magnetization [36,37]. This contribution can be significantly reduced by using a Carr-Purcell detection sequence [38,39], hence allowing the dipolar observation time window substantially extended. However, using multiple refocusing pulses for the detection spins requires the same number of pump pulses to be applied in the experiment. Thus, each refocusing pulse at the detection frequency has to be complemented with an inversion pump pulse. As seen in the 5-pulse PELDOR sequence in the previous section it is important that all spins are inverted by all pump pulses to avoid additional dipolar evolution time traces being detected as artefacts in the desired signal. This requires an inversion efficiency close to 100% within a certain pump pulse bandwidth, which must drop sharply to zero outside this frequency bandwidth. Due to its nonuniform inversion efficiency a rectangular pulse is not a good candidate for this task (see Figs. 1(c) and 2(j)). The adiabatic sech/tanh pulse fulfills these requirements much better (see Fig. 1(c)). A suitable pulse sequence with three shaped refocusing pulses is shown in Fig. 3(a). In this case, the shaped pulses are not designed to possess a large pump bandwidth but utilized because of their welldefined inversion profile, which matches the requirements specified above very well. The sech/tanh pulses used here are 400 ns long while having a bandwidth of only 50 MHz with b = 0.02/ns. The pulse sequence was applied at X- and Q-band frequencies on the spin labeled membrane transporter protein complex BetP (Fig. 3(b)), to investigate the symmetry properties of its trimeric state. The Q-band measurements have the advantage of intrinsically higher S/N but lacked somewhat in the performance of the sech/tanh pump pulse due to the restricted bandwidth of the Bru-

ker D2 resonator. Therefore, the inversion efficiency of the sech/tanh pulses deteriorated somewhat from the theoretical expected value close to one. The inversion efficiency is described by a parameter p, which provides the probability, averaged over the pulse bandwidth, that a spin is inverted twice by two consecutive pump pulses. Experimental measurements showed that p values between 0.7 and 0.75 for the 50 MHz bandwidth pulse defined above could be achieved used at Q-band frequencies with a Bruker D2 resonator and 0.9 when applied at X-band frequencies in both the Bruker MS-3 and MD-5 probe heads. Q-band measurements were carried out at 33.8 GHz with a 10 W solid-state amplifier and aqueous samples in 1.6 mm outer diameter quartz capillaries at 50 K. In the experiments the cavity resonance frequency was set to coincide with the center frequency of the sech/tanh pump pulse and the maximum of the nitroxide spectrum to ensure highest inversion efficiency. With p  0.75, a correction procedure can be applied to remove the unwanted dipolar evolution time traces (resulting from spin packets, which were not flipped by each pump pulse) from the desired signal before analysis of the dipolar trace. The only additional parameter, which has to be determined for the correction procedure, is the inversion efficiency p. After the signal is corrected for the intermolecular background, as described in [20], the intramolecular signal can be iteratively corrected to remove all unwanted dipolar evolution time traces arising from the nonperfect inversion efficiency p. The correction relies on the fact that all additional signals are also dipolar evolution time traces one would obtain by omitting one or two of the pump pulses. For three pump pulses (with two of them moving) five time-dependent pathways exist. The signal is a weighted sum of all these dipolar traces, which have different time zeros, determined by the time delays, and different time increments, given by the increment of the various pump pulses. Thus, if the dipolar evolution is known, all the pathways can be constructed by offsetting the evolutiontime zeros and rescaling the time axis accordingly. The weight W of each pathway is given by

W ¼ pN1 ð1  pÞ3N

ð6Þ

with N being the number of pump pulses involved in the respective dipolar pathway. Provided that the 7-pulse evolution is the largest time dependent contribution to the signal, the correction algorithm can use this relationship to successfully remove the unwanted components. In this case, the experimentally measured time trace is taken as a first approximation of the real dipolar evolution function. All the other unwanted pathways are then constructed from the measured signal according to their zero times, time increments and predicted weights, and subtracted from the raw 7-pulse time trace in a first step. The result of this procedure is taken as a better approximation to the true dipolar evolution function and used in place of the original experimental time trace in a second step. These steps are iteratively repeated until the resulting time trace does not change anymore and a self-consistent solution is reached. The 7-pulse CP-PELDOR method substantial prolongs the observation time window. Moving from the common 4-pulse sequence to the 7-pulse CP-PELDOR resulted in an almost doubling of the dipolar evolution time that could be measured for BetP (see Fig. 3(b)) in deuterated detergent solution (but protonated protein), from 6 to 11 ls as shown in Fig. 3(c) [20]. The extended observation time window provided by the CPPELDOR method is very promising in terms of applications, especially for membrane proteins, where the time window that can be achieved by 4-pulse PELDOR is often too short. It may be that other pulsed dipolar techniques, namely single frequency techniques such as SIFTER (single frequency technique for refocusing dipolar couplings) [40] and DQC-EPR (double quantum coherence)

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Fig. 3. (a) 7-pulse CP-PELDOR sequence with three sech/tanh pump pulses. The method was applied on the trimeric membrane transporter BetP (b). (c) Comparison of 4pulse PELDOR (gray) to 7-pulse CP-PELDOR (black). Both time traces are taken with the same accumulation time at Q-band frequencies (33.8 GHz). (e) The distance analysis obtained from the 7-pulse CP-PELDOR time trace (red) is in good agreement with predictions from the X-ray structure (blue). This figure is adapted from [20]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

[41], or alternative background correction procedures [22] can also take advantage of the Carr-Purcell pulse sequence and thus allow the experimental dipolar time window to be extended in these techniques too. However, it should be mentioned that the experimental challenges for such multi-pulse schemes are in general much higher than for the classical 4-pulse PELDOR experiment, which, when poorly adjusted, will not give results that include spurious frequencies, but only lower the S/N for the recorded data. The situation is different for the 7-pulse CP-PELDOR, where wrong adjustment of the pump pulses distorts the overall experimental signal very quickly to uncorrectable time traces. If p is too low, the iterative correction does not converge. The inversion efficiency parameter p has to be accurately determined experimentally [20]. Additionally, it is vital that the conditions of the experiment to determine p and the actual 7-pulse CP-PELDOR measurement are identical. In a 4-pulse PELDOR experiment with rectangular pulses, the Q-value of the MW resonator is usually chosen to be as high as possible while still allowing the rectangular pump and probe pulses of a given length to be generated in order to maximize the detection sensitivity. Using the same Q-value the resonator bandwidth with broadband shaped pulses might compromise the performance of the shaped pump pulses, leading to lower p values. This conflict might be avoided by using pump pulses optimized for the high Q-value of the resonator and off-resonance pumping to such a degree that the correction algorithm could be totally omitted. 3.3. Inversion filter with a broadband inversion pulse An approach to separate spectrally overlapping EPR signals from paramagnetic species with different longitudinal relaxation

times T1 is the usage of an inversion recovery filter [42]. This filter can be combined with hyperfine pulse sequences (REFINE: Relaxation Filtered Hyperfine Spectroscopy) [43], field-swept experiments [43] and PELDOR (iDEER: inversion recovery filtered DEER) [44]. For all these pulse sequences, the efficiency of the inversion recovery pulse is crucial. The experiment is performed by recording inversion recovery time traces over the field range B0 of the EPR species involved (see Fig. 4). An inverse Laplace transformation (iLT) of each inversion recovery time trace results in the separation of the different paramagnetic species, if the T1 relaxation times differ substantially. However, spectral diffusion processes, leading to non-exponential decay curves, diminishes this separation. Adiabatic inversion pulses can improve the performance of such an experiment, results of which are shown in Fig. 4. Spectral diffusion can be suppressed by using an adiabatic pulse for inversion recovery experiments [11]. Hence, the fast spectral diffusion component of an inversion recovery time trace can be substantially reduced. This allows more accurate T1e values to be found and correspondingly aids the separation of overlapping field-swept spectra [43]. 3.4. Fourier transformation of echo signals generated with broadband pulses Broadband pulses can also be used to create transversal magnetization, as described in Section 2. However, a phase dispersion occurs due to the different times within the broadband pulse when spins with different Larmor frequencies are effectively rotated (see Fig. 1(d)). This might result in zero macroscopic transversal magnetization after such broadband pulses. In order to generate a non-phase dispersed broadband Hahn echo, the BöhlenBodenhausen scheme [8] can be used. Here, the length of the p

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Fig. 4. Two-dimensional inversion-recovery field-swept spectrum of a mixture of BDPA (a,c-Bisdiphenylene-b-phenylallyl), TEMPO (2,2,6,6-Tetramethylpiperidine-1-oxyl) and CuHis (Copper-histidine complex) at X-band frequencies using an adiabatic inversion pulse (left). Contour plot of the amplitude values correlated to the relaxation rates after inverse Laplace transformation (middle). Individual slices taken from the contour plot (solid lines) and the field-swept spectra of the pure model compounds (dashed lines) are shown right. For comparison with data obtained with rectangular pulses see previous publications [45,46].

Fig. 5. (a) Broadband pulse schemes, based on the Böhlen-Bodenhausen concept to generate a Hahn echo, a refocused echo and a stimulated echo. The green and gray lines in (a) indicate the phase evolution of spins with different offset frequencies from the central frequency of the pulse. (b) The magnitude Fourier transformation of a broadband Hahn echo (green line), refocused echo (red line), and stimulated echo (cyan line) echo and, for comparison, the echo detected field-swept EPR spectra (blue line). The pulse duration used for the long pulses was 260 ns and 130 ns for the shorter pulses, both pulses had a bandwidth of 200 MHz. All experiments were performed on a 300 lM TEMPO solution in deuterated toluene and were measured at X-band frequencies at a temperature of 50 K. The video amplifier present in the commercial spectrometer was bypassed for these experiments. Figure adapted from [17]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

pulse needs to be half the length of the p/2 pulse. Vice versa, the frequency sweep rate b (see Eq. (4)) has to be roughly doubled for the p pulse in comparison to the p/2 pulse. In order to obtain the same bandwidth for the p and p/2 pulse the exact ratio needs to be simulated. The pulse scheme for a broadband Hahn echo [14], a refocused echo and a stimulated echo [17] with the corresponding Fourier transformed echo signals are illustrated in Fig. 5. A general expression of how to derive the relative pulse lengths for an arbitrary sequence by using the Böhlen-Bodenhausen scheme can be found here [31]. Based on this expression further EPR experiments using echos generated by chirped pulses were developed [15,47]. The magnitude spectra obtained by Fourier transformation of the resulting echo from all three broadband pulse sequences compare very well with the spectrum obtained by field-swept echo detection at X-band frequencies. The small differences at the edges of the spectra originate from inefficient pulse excitation at high offset frequencies. This problem can be compensated by taking the

resonator profile into account in the design of the broadband pulses [14]. Experimental results from our groups show that it is possible to still obtain complete spectral information using a sequence with broadband pulses all of the same length. Although this concept leads to smaller echo intensities (up to two times smaller in our cases) compared to the Böhlen-Bodenhausen schemes, it has also some significant advantages. In the BöhlenBodenhausen scheme if the p/2-pulse of the broadband Hahn echo sequence shown in Fig. 5 has a length of 260 ns, the optimum length for the p-pulse is only 130 ns, which may result in a pulse that is not adiabatic for the achievable B1max. By choosing all pulses to have the same length this allows the use of a longer p-pulse and consequently the range over which B1max for the p-pulse still results in an adiabatic pulse is larger. Consequently, the 260 ns adiabatic p-pulse is much more robust against distortions caused by the spectrometer hardware compared to the 130 ns p-pulse. For this reason, it might be beneficial to use pulses of a single length in the development of further broadband multi-pulse sequences.

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3.5. Broadband SIFTER The SIFTER experiment was introduced in 2000 as an alternative to the widely used PELDOR technique [40]. The most obvious difference is that PELDOR is performed with two microwave (mw) frequencies whereas SIFTER utilizes only one. Working on a single frequency can lead to higher S/N ratio on a SIFTER time trace compared to a PELDOR time trace as all the pulses can be set to the resonant frequency of the microwave cavity and therefore make use of the maximum microwave power at the sample whereas in PELDOR at least one of the frequencies must be positioned offresonance with the microwave resonator. The literature however only includes a small number of experimental SIFTER applications; this is due to two problems of the pulse sequence. Firstly, the inversion profile of rectangular pulses (see Fig. 1(c)) leads to the appearance of artifacts in SIFTER time traces. The artifacts arise because the probability of spins to be flipped by both p-pulses decreases strongly at larger offset frequencies. This is the same problem also witnessed in the CP-PELDOR experiment (see Section 3.2). By using broadband pulses [17], or spin probes narrow in spectral width [48], those artifacts can be suppressed completely. In addition, the possibility to construct broadband pulses with a bandwidth larger than the spectral width of nitroxides enables a modulation depth of up to 95% at X-band frequencies. A similar value was achieved at Q-band frequencies using chirp pulses [21]. The second problem is the ambiguity in the definition of the background function for single frequency EPR techniques. Whereas in PELDOR experiments, where the time delays between the observer pulses are fixed, resulting in a background function that only depends on the intermolecular dipolar coupling Binter(t), in SIFTER, due to the movement of the observer pulses relative to each other, the background function contains an additional term Brelax(t) resulting from relaxation. The function of the observed signal VSIFTER(t) in SIFTER is therefore:

V SIFTER ðtÞ ¼ Brelax ðtÞBinter ðtÞFðtÞ

ð7Þ

F(t) is called the form factor, which contains the intramolecular dipolar information. Brelax(t) originates from the transversal relaxation of the electron spin, mostly due to hyperfine coupled nuclei [36]. This additional background function can be measured using a 3-pulse sequence (see Fig. 6(b)) with the same experimental settings as for the SIFTER sequence (see Fig. 6(a)) with the second p/2 pulse removed. A division of the SIFTER time trace by the 3-pulse data results in a time trace free of Brelax(t). Subsequently Binter(t) can be removed by fitting an exponential function to the end of the time trace, analogous to the standard PELDOR background correction procedure. The results of such a background correction procedure are shown in Fig. 6. Here, broadband SIFTER and the 3-pulse sequence are measured on doubly spin-labeled POTRA domains [49] at Q-band frequencies. The normalized intramolecular dipolar evolution functions obtained using both SIFTER and PELDOR experiments are very similar after the background correction procedure. One drawback of this procedure is that two experiments are necessary. However, the better S/N of SIFTER compensates for the additional experimental time. The reliability and correctness of this background correction procedure is the subject of further investigations. 4. Design of shaped pulses for EPR spin systems based on optimal control methods So far, the majority of applications of shaped pulses in EPR have focused on families of frequency-swept (adiabatic or nonadiabatic) pulses with a strictly monotonous frequency ramp as a

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function of time. This subset of pulses can increase the performance of EPR experiments relative to simple rectangular pulses and each family of frequency-swept pulses, such as sech/tanh and WURST, depends on a relatively small set of parameters (see Section 2) which can be optimized numerically or even experimentally. However, they only constitute a vanishingly small subset of all possible pulse shapes that can be created by state-of-the-art AWGs. In this context, important questions of theoretical and practical interest are still open and are appropriate to consider in the context of the current perspective article: Are monotonic frequency-swept pulses optimal or do more general pulse shapes exist that have improved performance in real EPR experiments, and how could they be found in practice? In the following we consider this question from the point of view of pulse design based on the principles of optimal control theory. The design of optimal pulses for real EPR samples and practical applications is a highly non-trivial task. In general it is necessary to take into account inhomogeneous distributions of spin system parameters (offset frequencies, electron-electron couplings, hyperfine couplings, etc.), ranges of relaxation rates (transverse, longitudinal, multi-quantum, etc.), experimental constraints (maximum pulse amplitudes due to amplifier constraints and/or maximum average power of a pulse sequence to limit heating effects) and experimental imperfections (receiver dead time, B1 inhomogeneity, amplitude and phase transients due to linear as well as nonlinear pulse distortions e.g. created by finite resonator bandwidths and non-ideal amplifiers, up-converters or multipliers). This task can be viewed as a constrained optimization problem of an ensemble of dynamical systems. This is a topic of OCT [51– 53], which provides both analytical and highly efficient numerical methods to solve such problems. OCT applications range from motion control in robotics and the control of space probe trajectories to the control of the dynamics of quantum systems using electromagnetic fields [54–57]. In the following, we will focus on OCT results relevant for EPR spectroscopy. However, if not specifically stated otherwise, the presented results apply to generic spin systems representing electron spins as well as nuclear spins. Analytical optimal control theory, which is also known as geometric OCT, is typically limited to relatively simple model systems. Nevertheless, OCT has already provided a large number of analytical solutions for the globally optimal pulse sequence in highly non-trivial cases, providing not only the physical performance limit but also a complete, in-depth understanding of the control problem and of its solution. Examples include time-optimal control problems of state-to-state transfer [58,59] and universal rotation pulses [60] as well as relaxation-optimized control problems [61–64] for individual and coupled spins [65,66]. It is interesting to note that in geometric OCT, the globally optimal solutions are often primarily defined in terms of specific, well-characterized optimal trajectories that lead from the initial to the target state, whereas the actual optimal pulse shapes are only secondary, i.e. they can be derived from the optimal trajectories [61–63]. Unfortunately, currently available analytical approaches are not yet able to include all aspects of realistic experiments. Fortunately, analytical OCT methods are complemented by numerical OCT algorithms, which can essentially be applied in all experimental settings that can be accurately and efficiently simulated. In cases where analytical optimal control solutions are known, the same solutions were also found independently using numerical optimal control methods [55], suggesting that in more complicated cases numerical OCT tools can also be used to explore the physical limits of pulse sequence performances. Although in general there is no guarantee that a numerically optimized pulse has reached a global optimum, such a pulse may still be highly valuable if it actually provides substantial gains in experimental performance compared to previous approaches [67–71]. Although numerical OCT algorithms provide

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Fig. 6. (a) Broadband SIFTER sequence. In contrast to [17] the pulse length is identical for all pulses. (b) Broadband 3-pulse sequence to measure the additional background function Brelax(t). (c) Broadband SIFTER (black line) and 3-pulse sequence (green line) measured on nitroxide spin-labeled POTRA domains [49] at Q-band frequencies and 50 K. For better comparison, the 3-pulse time trace was vertically scaled to the last point of the SIFTER time trace. The parameters for the pulses were set to tp = 300 ns, n = 5, b = 0.0024 ns1 and 0.5BW = 1.7 rad ns1. (d) Background-corrected PELDOR [49] (gray line) and broadband SIFTER (navy line) time traces, scaled to the same modulation depth. For PELDOR the standard background correction procedure was applied using DeerAnalysis [50]. For SIFTER the raw time trace (c) was divided by the 3-pulse sequence time trace (c) afterwards which the standard background correction procedure was applied. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

only limited direct insight into the underlying principles of the optimal solutions, in several cases numerical solutions that demonstrated unexpected performance and characteristic features of the corresponding pulse shapes have triggered successful quests for analytical OCT solutions. Hence, numerical approaches have formed an important stepping-stone to a fundamental understanding of the underlying principles [72,73]. In contrast to simple optimization algorithms, which are often limited to a few dozens of pulse sequence parameters, optimalcontrol-based algorithms such as the GRAPE (gradient ascent pulse engineering) approach [55] can efficiently optimize thousands and even hundreds of thousands of parameters [69]. Numerical OCT algorithms have been implemented in several software packages, including SIMPSON [74], SPINACH [75,76], and DYNAMO [77], which are widely used in the magnetic resonance and the quantum control community. In practice, a pulse created by an AWG is defined and completely characterized by the amplitudes and phases during all individual time slices into which the pulse is partitioned. For example, a pulse with a duration of one microsecond - sliced in segments of one nanosecond - is characterized by a series of 1000 individual amplitudes and phases. The large number of tractable optimization parameters makes it possible to optimize all these individual pulse parameters without any a priori restriction to a specific class of pulse families, such as composite or adiabatic pulses. With this approach, detailed systematic studies have been performed for broadband pulses with bandwidths much larger than the achievable Rabi frequency of the maximum pulse amplitude, providing empirical scaling relations that make it possible to consider trade-offs between different parameters, such as bandwidth, range of B1 robustness and pulse duration, even before optimizing pulses [78–80]. Typically, the pulse duration only scales linearly with the desired offset range if the resonator bandwidth is much larger than

the offset range. In this context, it is helpful to distinguish different classes of individually optimized pulses based on their function in a pulse sequence and their scaling relations: Universal rotation (UR) pulses [80,81] with a defined rotation axis and rotation angle (e.g. for refocusing) and point-to-point (PP) pulses [78,79,81], which rotate a magnetization vector from a specific initial point on the Bloch sphere as closely as possible to a target point, e.g. from the z axis to the x axis for excitation pulses or from z to z for inversion pulses. Depending on the task and the duration of broadband pulses, very different pulse families are found. For pulses with durations on the order of a 360° rectangular pulse or shorter, composite pulses with constant amplitude and discrete phase switching between periods of constant pulse phase are optimal, whereas optimal pulses for longer durations typically have continuous phase modulations [78,80]. In general, a phase modulation (PM) function can always be translated into frequency modulation (FM) function. For relatively simple, smooth PM functions, it is often possible to qualitatively and even quantitatively understand pulse properties based on the concept that the instantaneous excitation frequency is the corresponding value of the FM functions. However, in addition to a monotonic frequency sweep through the desired range of offset frequencies, the numerically found OCT inversion pulses also have periods at the beginning and at the end, where the sweep direction is reversed, i.e. improved broadband inversion is found, if the pulses are not required to have a strictly monotonic frequency sweep for the entire pulse duration [78]. For more demanding applications, such as excitation and refocusing pulses, OCT pulses provide significant gains compared to conventional approaches and often have a much more complicated PM function. This applies to individually optimized pulses as well as to cooperatively optimized excitation and inversion pulses (vide infra). A time-frequency analysis of these high-

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performance OCT pulses in terms of their spectrogram representation [82] reveals that at each point in time, these pulses often have not a single but several simultaneous irradiation frequencies. Hence, in contrast to the case of frequency-swept pulses, the translation of the PM function into a single FM function is in general not sufficient to provide an intuitive understanding of pulse properties, such as the offset-dependence of the phase of transverse magnetization created by an excitation pulse, in terms of a sharp instantaneous irradiation frequency. Whereas in liquid-state NMR, B1 inhomogeneity is the only experimental imperfection that needs to be taken into account in order to achieve an excellent match between simulated and experimental pulse performance, in typical X- and Q-band frequency EPR experiments, amplitude and phase transients due to limited resonator bandwidths must also be included. As discussed in Section 5 these transient effects can be characterized by an impulse response function [83], which can be experimentally measured [10]. This makes it possible to calculate for any input pulse shape to the resonator the corresponding effective pulse shape inside the resonator seen by the spins. Conversely, for any effective pulse shape, the corresponding input pulse shape can be calculated. However, such simple pre-compensated pulses typically require significantly larger pulse amplitudes to compensate for the transient effects, which may exceed the usable range of the amplifier. A more efficient strategy is to take the impulse transfer function into account already during the optimization of the input pulse shape [10] such that its maximum amplitude is within the specifications of the amplifier and the effective pulse shape effects the desired spin dynamics. This is illustrated in Fig. 7 for a prefocused broadband excitation pulse (BEBOP) for EPR optimized such that 200 ns after the end of the pulse the transverse magnetization is aligned along the x axis (pulse duration: 1 ls, maximum pulse amplitude cB1/2p = 12 MHz, desired excitation bandwidth: 80 MHz). The experimentally measured impulse transfer function was taken into account in the optimization, but effects of hyperfine couplings and relaxation were neglected in the pulse optimization. Fig. 7 compares the simulated spectra for uncoupled spins and the experimentally measured spectra of a fluoranthenyl sample as a function of the offset frequency for three pulse sequences with identical maximum B1 amplitudes: A simple Hahn echo (left column) and broadband BEBOP pulses optimized omitting (middle column) and including (right column) the impulse response. The first row shows the ideal input pulses to the resonator, whereas the second row shows the effective field interacting with the spins inside the resonator. The offset-dependent x magnetization created 200 ns after the end of the pulse shown in the third row by gray curves for the ideal input pulse and by black curves for the effective field inside the resonator, respectively. In the fourth row, the simulated black curves of the third row are scaled according to the measured offset-dependent signal distortions of the acquisition process and the gray curves represent experimental signals. A reasonable match between simulated and experimental signal amplitudes is found. Scaling up the B1 amplitude of the pulses results in a corresponding increase of the excitation bandwidth if the pulse durations are reduced by the inverse scaling factor of the pulse amplitude. As shown in [84], it is also possible to take nonlinear pulse distortions into account in the OCT algorithm. In addition to broadband pulses, band-selective excitation, inversion and refocusing pulses with sharp transitions between the pass-bands and stop-bands are highly desirable in many EPR experiments, such as CP-PELDOR [20]. OCT methods have been used to optimize band-selective pulses in NMR settings [54,85,86] to achieve the most narrow transition range for a given pulse duration and to have a negligible overall effect outside of the pass band. This was achieved by a combination of bandpass-

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filtered pulse shapes in the optimization and by giving the critical spectral ranges a higher weight in the definition of the figure of merit in the optimizations. Finally, significant improvement of pulse performance for broadband or selective applications can be obtained by the concept of simultaneously optimized cooperative pulses [87,88]. Most experiments do not consist of only a single pulse, but of highly orchestrated sequences of pulses that are separated by delays, which are either constant or varied in a systematic way. This provides additional opportunities to improve the overall performance of experiments beyond what is achievable by simply combining the best possible individually optimized (composite or shaped) pulses. The cooperativity of such pulses provides important additional degrees of freedom in the pulse sequence optimization, because the individual pulses do not need to be perfect if they compensate for each other’s imperfections. The analysis and systematic optimization of cooperative effects between different pulses results in a better overall performance of a pulse sequence and shorter pulse durations. Cooperativity effects can be exploited between corresponding pulses in different scans of a multi-scan experiment (ms-COOP pulses) [87] and for pulses in the same scan (s2-COOP) [88]. Fig. 8 shows the dramatic improvement of pulse sequence performances for the example of broadband Ramsey-type experiments consisting of two p/2 pulses, separated by a delay s. Note that an ideal Ramsey sequence creates a modulation of the longitudinal magnetization component of the form cos(xs), where x is the offset of the spin. The Ramsey sequence is used as a frequency-labeling element in many two-dimensional experiments and also in stimulated echo sequences (consisting of three p/2 pulses), where the first two pulses form a Ramsey building block. The performance index U of an s2-COOP Ramsey pulse pairs is defined as 1 minus the root-mean-square difference between the simulated and the ideal modulation cos(xs) of the longitudinal magnetization component as a function of offset. Excellent overall performance can be achieved using a pair of short, simultaneously optimized shaped pulses (s2-COOP0.6) that are only three times longer than a simple rectangular p/2 pulse. The subscript 0.6 indicates that each of the pulses incudes an effective offset-evolution period of 60% of the pulse duration [88], which is similar to rectangular p/2 pulses, which have an effective offset-evolution period of about 63% of the pulse duration. In order to reach the same performance of the Ramsey experiment, much longer individually optimized PP or UR pulses would be required, as shown in Fig. 8. We also have developed cooperative pulses for Hahn echo sequences consisting of an excitation and a refocusing pulse. In contrast to pre-focused pulses described previously (see Fig. 7), the cooperative echo pulses are not designed for a fixed effective evolution period, but to work for arbitrary separations between the two pulses. Fig. 9 shows spectrogram representations [82] of a broadband echo sequence based on chirped pulses [8] and of a pair of significantly shorter cooperative pulses optimized for a bandwidth of 100 MHz and maximum B1 amplitude of only 30 MHz. The performance of these two approaches (and of a standard Hahn echo using rectangular pulses) is shown in Fig. 10 (without and with the application of the EXORCYCLE [8,89] four-step phase cycle). The simulated echo amplitudes and echo phases illustrate that a combination of two frequency-swept pulses can indeed generate a large excitation bandwidth. However, as previously pointed out [90], the phase of the resulting echo is very sensitive to B1 inhomogeneity, which results in reduced echo amplitudes. In Fig. 10(b) and (b0 ) this B1-dependent phase of the echo is depicted by the change in color. In contrast, the phase of the echoes created by the optimized cooperative pulses is constant for the

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Fig. 7. Comparison of conventional and optimized pulses designed to acquire the signal 200 ns after the end of the last pulse and their offset profiles. The left column represents a conventional Hahn echo sequence based on rectangular p/2 and p pulses. The middle and right columns represent results of optimizations without (middle column) and with (right column) transient effects taken into account in the optimization. The panels in the first and second rows show the x- (red line) and y-(blue line) components of simulated input (a, a0 , a00 ) and effective pulses (b, b0 , b00 ), respectively. The third row (c, c0 , c00 ) shows simulations of the offset-dependent x magnetization created 200 ns after the end of the pulse for the input pulse (gray curves) and for the effective pulse (black curves). Finally, the black curves in the fourth row (d, d0 , d00 ) represent the simulated x magnetization created 200 ns after the end of the pulse of the output pulses multiplied by the measured frequency response of the video amplifier. In panels d0 and d00 , gray curves represent experimental spectra acquired for a small single crystal (0.5  0.3  0.3 mm) of the organic conductor Fluoranthenylhexafluorophosphate ((Fa)+2 PF 6 ). This figure is adapted from [10]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

optimized range of ±15% of B1 scaling (as seen in Fig. 10(c) and (c0 )). This results in a larger overall echo amplitude of the COOP echo sequence, compared to both WURST and rectangular pulses, due

to the coherent superposition of the individual echo contributions even if relaxation effects are neglected. In applications with short transverse relaxation times (comparable to or shorter than the

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Fig. 8. Performance U of different families of Ramsey sequences as a function of the duration T of the (shaped) p/2 pulses for the case of a maximum B1 amplitude of 30 MHz and a seven times larger bandwidth of 210 MHz. In the optimizations and simulations, uncoupled spins are considered. The open square, labeled ‘‘rect”, corresponds to rectangular p/2 pulses with a duration of 8.3 ns. The red triangles labeled s2-COOP0.6 correspond to optimal same-scan cooperative p/2 pulses. The subscripts indicate the fraction of each pulse duration during which an effective offset evolution takes place. Note the significantly shorter duration of s2-COOP0.6 pulses compared to individually optimized PP and UR pulses with the same performance U (figure adapted from [88] by rescaling the frequency range by a factor of 3000 and the time by a factor 1/3000 to bring the pulse from the realm of NMR to the realm of EPR). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

pulse duration) even larger gains are expected due to the significantly shorter overall pulse duration (100 ns + 100 ns = 200 ns for the COOP echo sequence of Fig. 9(b) vs. 256 ns + 128 ns = 384 ns for the WURST-based echo sequence of Fig. 9(a)). As shown previously [91], the radio-frequency amplitude-dependent phase errors of the Böhlen-Bodenhausen scheme are due to an offset-dependent dynamical Bloch-Siegert phase, which can largely be compensated using an additional frequency-swept p pulse. However, this results in a further increase of the overall pulse duration, which can offset any gains from refocusing of the dynamic phase shift in EPR applications of fast relaxing spins, as shown experimentally [31]. For practical applications with realistic samples, typical relaxation rates can be considered in the OCT algorithm to find relaxation-optimized pulses that provide the best compromise between excitation efficiency and relaxation losses [92]. BEBOP pulses optimized for uncoupled spins will in general not work with the same fidelity for coupled spin systems if the pulse duration is not much shorter than the inverse of the coupling constant. However, as demonstrated in the case of NMR and DNP applications [50,93–98], coupling effects can also be taken into account in the OCT algorithm and it is expected that broadband EPR pulses can be optimized that are robust to a large range of scalar hyperfine couplings (to be published elsewhere). In this case, the expected range of the coupling constant has to be specified in the optimizations. As demonstrated for single crystals, numerical OCT optimization of shaped pulses can also be performed for spin systems with anisotropic pseudo-secular hyperfine couplings [99]. In this case, the extension to powders or frozen solutions would require the optimization for a large number of crystallite orientations if the hyperfine coupling tensor is known or can be estimated. If this is not the case, it would also be possible to consider a reasonable range of expected coupling tensors in the pulse optimization. As for any sufficiently large spin system and/or sufficiently large number of robustness constraints, the computational cost to simulate all the effects of a pulse during an optimization scales exponentially. For this reason, it would be necessary to find the most

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simple spin system model that captures the essential dynamics of the EPR spin system. However, based on experiences from the optimization of pulse shapes for solid-state NMR experiments of powder spectra (with the additional complication of magic angle sample spinning) it is expected that it may also be possible to develop robust OCT-based pulses and pulse sequences with optimized performance for pulsed EPR experiments in the foreseeable future. One approach to speed-up the search of optimal control solutions is to minimize the number of optimization parameters, i.e. to reduce the parameter space. This can be a good approach, if preliminary studies based on unconstrained optimizations have provided strong indications about promising pulse families. Commonly used parameterizations of smooth functions (representing pulse amplitudes, pulse phases or pulse frequencies) include Fourier series [100], Gaussian pulse cascades [101] and cubic spline interpolation of a small set of anchor points [102] and efficient OCT approaches have been developed for optimizations in such reduced parameter spaces [103]. In addition, variations of OCT algorithms were developed for experimental settings, where the spectrometer can only switch between a relatively small number of discrete pulse amplitudes or pulse phases [104].

5. Pulse distortions by the microwave resonator EPR spectrometers use microwave resonators wherein the sample interacts with the B1 field. These serve several very important functions: At first, they spatially separate the electrical and magnetic field components to ensure that the sample does not absorb the electric field and only the magnetic field penetrates the sample. Secondly, it enhances the B1 field at the location of the sample due to its ability to store energy. A resonator can be regarded as a LC circuit where the electric field is stored in the capacitor and the magnetic field in the coil, where it interacts with the sample. LC circuits are resonance structures whose frequency behavior is described by a Lorentzian curve. Resonance structures have a non-uniform phase and amplitude response. That means the B1 field enhancement is hence frequency dependent and the phase of the B1 field inside the resonator alters with respect to the incident microwave. These effects are linear and time-invariant because the output of an LC circuit is always proportional to the input and does not have a memory. The situation will become more complex if, for example, the resonator heats up and electrical losses are increased. The linear effects can be quantified by linear response theory, which states that the output pulse shape can be calculated if the impulse response function of the system is known. The knowledge of the impulse response function enables us to design pulses in such a way that they have exactly the desired amplitude and phase within the resonator. However, it is not so easy to measure this function directly as it is impossible to generate a delta shaped excitation pulse and measure the response directly inside the resonator. It is well known from linear response theory that a pseudo-stochastic maximum length binary sequence can be used to obtain the impulse response function by deconvolution [105]. This approach has also been used in the field of NMR [2,6] and EPR [10,106]. Fig. 11 shows the in-phase and quadrature components of the signal measured with an external field probe upon excitation with such a maximum length sequence (phase adjusted for the in-phase channel). The exact procedure of the measurement is described in [10]. The signal was recorded with a field pick-up coil placed in the stray field of the resonator. This setup was not optimized in two aspects: First, the pick-up coil was arranged above the resonator thus

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Fig. 9. Representations in terms of x and y amplitudes mx(t) (red lines) und my(t) (blue lines) (top row) and spectrogram representation [82] (bottom row) of broadband echo sequences designed to cover a bandwidth of 100 MHz with a maximum nominal B1 amplitude of 30 MHz. (a, a0 ) chirp pulses based on the Böhlen-Bodenhausen scheme [8]. The first excitation pulse has a length of 240 ns with reduced maximum amplitude of 7.5 MHz whereas the refocusing pulse has a length of 120 ns and a full amplitude of 30 MHz. (b, b0 ) depicts a COOP-echo sequence designed to be robust with respect to a B1 scaling in the range between 0.85 and 1.15. The excitation pulse and the refocusing pulse both have durations of only 100 ns. Arbitrary delays between excitation and refocusing pulses were chosen in order to clearly separate the spectrograms of the individual pulses. For a comparison of the resulting simulated amplitudes and phases of transverse magnetization as a function of offset and B1 scaling, see Fig. 10. The relative amplitude of the spectrograms are shown, normalized to the maximum amplitude in (a0 ). For (a0 ) and (b0 ), the same Gaussian window function with a full width at half height of 40 ns was used [82]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Simulated amplitude (represented by brightness) and phase (represented by color: red = +x, yellow = +y, green = x, blue = y) of transverse magnetization at the time of the echo as a function of offset frequency m and B1 scaling relative to the nominal amplitude of B1,nom = 30 MHz. The left panels (a and a0 ) show the simulated performance of a standard echo sequence consisting of rectangular p/2 and p pulses. The middle panels (b and b0 ) show the performance of WURST pulses based on the Böhlen-Bodenhausen scheme [8], c.f. Fig. 9(a) and (a0 ) and the right panels (c and c0 ) show the performance of the COOP-echo sequence of Fig. 9(b) and (b0 ) optimized for a bandwidth of 100 MHz with B1,nom corresponding to a nutation frequency of 30 MHz and scaling factors B1/B1,nom ranging from 0.85 to 1.15. In all cases the top row (panels a– c) presents the results for a single echo experiment with no phase cycle, whereas the bottom row (panels a0 -c0 ) shows the results of four repetitions of each echo experiment with different phases according to the EXORCYCLE scheme [8,89]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 11. Excerpt of the quadrature response upon a maximum length binary sequence excitation (black) recorded with a pick-up coil in the stray field outside of the microwave resonator. The in-phase and quadrature response are shown in blue and red lines, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

blocking the sample access. Therefore, the response function was only determined without the sample inserted, whereas the real measurement was performed without the pick-up coil in place. Second, the field was not measured right at the position of the sample inside the resonator. The assumption may be valid that the stray field outside the resonator is proportional to the field generated inside where the sample is located. However, additional fields are present at the sample position. These disturbances may be large enough to spoil the effort made to alter the pulse shapes in order to correct for the cavity response. It was shown successfully that it is possible to measure the magnitude B1 profile by means of a nutation experiment over the bandwidth of interest. This is advantageous since the B1 amplitude is measured directly where the spins are located. However, the phase information cannot be measured directly and has to be calculated from the magnitude response by means of the Bode relation and assuming that no all-pass filters are present [11]. Han et al. measured the spin response for small tip angles to extract the linear distortions of the signal transducing pathway [107]. A better procedure to reliably take the impulse response into account would be to build a probe head with an integrated B1 probe located in such a way that it does not hinder inserting a sample into the cavity. The probe head should be designed in such a way that the B1 probe detects only the B1 field created by the resonator and not the standing waves in the microwave transmission lines. With such a device a pulse optimization could be done in a very precise manner with potentially significant benefits for most of the pulses and applications described in this article.

6. Conclusions and outlook The opportunity to manipulate electron spin systems with pulses modulated in amplitude and phase on a nanosecond time scale offers new exciting opportunities for EPR pulse sequences and applications. The use of adiabatic broadband inversion pulses and simple frequency-swept pulses for transversal magnetization has already demonstrated very promising improvements compared to pulse sequences based on rectangular pulses alone, and allowed the extension of existing EPR pulse sequences and the development of new sequences. Here we have illustrated this aspect primarily with our own applications of such pulses to dipolar EPR spectroscopy: We have shown that the modulation depth for PELDOR performed on species with large spectral widths can

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be improved through the application of broadband adiabatic inversion pulses. This also leads to better spectral separation for inversion-recovery filter applications. The high selectivity of sech/tanh inversion pulses allowed the application of Carr-Purcell refocusing schemes for PELDOR experiments, substantially prolonging the dipolar evolution time-window that can be observed. We also included the first demonstration of such a scheme to improve the performance of the SIFTER pulse sequence for dipolar spectroscopy and we proposed a background correction procedure for this method. Collectively these examples give a glimpse of the new potential that AWG shaped pulses offer to EPR spectroscopy. In order to take full advantage of the flexibility of AWGs, the ultimate question is: What is the best possible experiment to obtain the desired information in a given application? This deceptively simple question is in fact very profound with many different facets. Optimal control theory appears to provide a general framework to address them in a systematic way. OCT requires an accurate model to simulate the spin dynamics in a realistic setting, taking into account experimental limitations and imperfections. Furthermore, a figure of merit needs to be defined that makes it possible to quantify performance, which in itself is a non-trivial task and has a strong influence on the usefulness of the resulting pulse sequences. Traditionally, the focus was on the design and optimization of individual pulses that are subsequently combined to construct highly orchestrated multi-dimensional pulse sequences. Only recently it was demonstrated that the concurrent optimization of so-called cooperative pulses [87,88] can lead to significantly improved overall pulse-sequence performance. Furthermore, novel OCT-based approaches for the optimization of the sensitivity per unit time [108] have the potential to maximize the information that can be obtained in a given time period. An important long-term goal is to optimize entire pulse sequences for realistic settings. A critical point is whether or not the complexity of the optimization task remains tractable for real applications. Even if it turns out that this is not currently possible for all applications, constant improvements of computing power in combination with ongoing developments toward more powerful analytical and numerical OCT methods provides an optimistic long-term perspective. It would be highly desirable to be able to optimize or re-optimize pulse-sequences on the fly, e.g. to adapt the experiment to sample-dependent resonator Q-factors. Furthermore, the combination of numerical optimizations with experimental feedback strategies has the potential to further improve pulse sequence performance [57]. A close collaboration between experimental and theoretical groups will be crucial to fully explore the scope of what is currently possible with the available hardware. The first FT-EPR applications of OCT-derived pulses on organic radicals were very successful, thus demonstrating that some of the concepts developed in the field of NMR can also be applied to EPR. Of course, the instrumentation of pulsed EPR that was developed and optimized over many years for monochromatic rectangular pulses has to be modified to optimally suit such pulses. Therefore, at high magnetic fields (>3 T), where the available mw excitation power is still rather limited and bandwidth of the mw resonator by far exceeds the available B1 field strength, broadband pulses might become especially important. For lower mw frequencies, new types of resonance structures, such as micro-resonators, which are under development for very small sample sizes [109], might be interesting for EPR applications. Improving the homogeneous performance of such shaped pulses within a given bandwidth might also allow the development of new multidimensional experiments in EPR, similar to those currently used in NMR. In conclusion, the application of shaped pulses in EPR is just in its infancy and many interesting and new applications can be expected in the future.

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P.E. Spindler et al. / Journal of Magnetic Resonance 280 (2017) 30–45

Acknowledgments We thank the priority program SPP1601 ‘‘New Frontiers in EPR Sensitivity” (Grant Pr 294/15-2 and Gl 203/7-2) for financial support. We gratefully acknowledge the contribution of Dr. Alice Bowen and Dr. Burkhard Endeward and thank Silke Schneider for technical assistance in manuscript correction.

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