Pulses for 1.2 GHz spectrometers

The problem

Sensitivity in NMR rises steeply with magnetic field, which is why the community spent thirty years and a great deal of money getting from 500 MHz to 1.2 GHz (28.2 T). But field strength scales up something else too: the chemical shift range, measured in hertz, grows in direct proportion. A ¹³C spectrum that spanned 30 kHz at 600 MHz spans twice that at 1.2 GHz.

Meanwhile the radiofrequency amplitude a probe can deliver has not doubled. It is limited by arcing, by coil heating, and — for biological samples in salty buffer — by how much power you can deposit before you cook the sample. So the pulse has to cover twice the bandwidth with roughly the same power budget.

There is a third constraint that is easy to forget. The B₁ field is not uniform across the active volume of the coil. Spins at the edge of the sample see a weaker pulse than spins at the center — often 20% weaker or more. A pulse calibrated perfectly for the center is miscalibrated everywhere else, and the signal you lose is gone for good.

• Broadband — must invert or excite uniformly across a much wider spectrum.
• Low power — must fit inside probe and sample-heating limits.
• Robust — must work despite 20%+ variation in B₁ across the sample.

Conventional shapes let you buy any two of these. A hard rectangular pulse is broadband but demands high power and is exquisitely sensitive to miscalibration. An adiabatic pulse is beautifully robust to B₁ variation but needs a long duration and substantial power to satisfy the adiabatic condition. Selective shaped pulses are gentle but narrow. The compromise is structural, not a failure of craftsmanship.

The approach

Optimal control refuses the compromise by refusing the premise that a pulse should have a name. Rather than choosing from a catalog of analytical shapes, the pulse is discretized into a few hundred to a few thousand time slices, each with its own amplitude and phase. That is the parameter vector. A gradient-based optimizer — GRAPE and its descendants — then maximizes a fidelity function that is evaluated not at one operating point but averaged over an ensemble: every resonance offset across the target bandwidth, crossed with every B₁ scaling factor present in the coil.

This is the part that matters. A pulse optimized at a single offset and nominal power is not robust — it is merely optimal somewhere. By putting the whole distribution of operating conditions inside the cost function, robustness stops being a property you hope for and becomes a property you optimized for. Peak amplitude enters as a hard constraint, so the result is admissible on real hardware by construction rather than by luck.

What came of it

The resulting pulse library for 1.2 GHz instruments was published in Science Advances in 2023, and was recognized with the Ernst Award for best paper from the Division of Magnetic Resonance of the German Chemical Society (GDCh), and the JMR/JMRO Young Scientist Award at EUROMAR 2024.

A follow-up in the Journal of Magnetic Resonance (2026) extended the approach to low-power sequences for a 5 mm triple-resonance cryogenic probe optimized for proton detection — the configuration most biomolecular groups actually run. Working with a probe manufacturer meant the constraints in the cost function were the real ones.

The pulses are usable output, not just a result. They exist as pulse programs that can be loaded on a spectrometer, and the optimizer that produced them is open source — see Pulsar.jl.