pulse(Par)
pulse(Par,Opt)
[t,IQ] = pulse(...)
[t,IQ,modulation] = pulse(...)

Pulse length, in microseconds.

Pulse shape name in a character array structured as 'AM/FM' (or just 'AM'), where AM refers to the amplitude modulation function and FM to the frequency modulation function.

The available amplitude modulation functions are 'rectangular', 'gaussian', 'sinc', 'halfsin', 'quartersin', 'tanh2', 'sech', 'WURST', 'GaussianCascade', 'FourierSeries'. Different AM functions can be multiplied by concatenating the keywords as 'AM1*AM2/FM'.

The available options for frequency modulation are 'none', 'linear', 'tanh', 'uniformQ'. If only an AM keyword is given, the FM function is set to 'none'.
The 'uniformQ' option can be used to obtain the frequency modulation corresponding to a pulse with offset-independent adiabaticity given a specific amplitude modulation function (see Tannús, A., Garwood, M. Improved Performance of Frequency-Swept Pulses Using Offset-Independent Adiabaticity. J. Magn. Reson. A 120, 133ÿ137 (1996) (DOI: 10.1006/jmra.1996.0110 )), for example for higher-order sech (HSn) pulses.

Here are a few examples:

Par.Type = 'sech/tanh';     % hyperbolic-secant pulse
Par.Type = 'WURST/linear';  % linear chirp pulse with WURST amplitude envelope
Par.Type = 'gaussian';      % constant-frequency gaussian-amplitude pulse

Additional parameters are required to fully specify the various AM and FM functions, see below.

Pulse flip angle, in radians. For example, Par.Flip = pi specifies a 180-degree pulse.

Flip is used to calculate the pulse amplitude. For amplitude-modulated pulses, it is calculated via the integral of the amplitude modulation function. For frequency-modulated pulses, the procedure described in Jeschke, G., Pribitzer, S., Doll, A. Coherence Transfer by Passage Pulses in Electron Paramagnetic Resonance Spectroscopy. J. Phys. Chem. B 119, 13570ÿ13582 (2015) (DOI: 10.1021/acs.jpcb.5b02964) is used.

Pulse amplitude, in MHz. If Par.Flip is given, Par.Amplitude is ignored.

Critical adiabaticity for the calculation of pulse amplitudes for frequency-swept pulses. If Par.Qcrit is given, Par.Amplitude and Par.Flip are ignored.

The pulse amplitude is calculated from the critical adiabaticity as described in Jeschke, G., Pribitzer, S., Doll, A. Coherence Transfer by Passage Pulses in Electron Paramagnetic Resonance Spectroscopy. J. Phys. Chem. B 119, 13570ÿ13582 (2015) (DOI: 10.1021/acs.jpcb.5b02964).

Pulse frequency, in MHz (default = 0). For frequency-swept pulses the start and end frequencies are required.

Par.Frequency = -10;  % constant frequency offset
Par.Frequency = [-50 50];  % up-sweep
Par.Frequency = [50 20];  % down-sweep

Overall pulse phase, in radians, e.g. 0 for +x, pi/2 for +y, etc. If not given, it defaults to 0.

Time step for the definition of the waveform, in microseconds. If not given, it is automatically determined based on the Nyquist sampling criterion and the input pulse parameters.

Depending on the AM function, additional fields need to be specified:

• 'rectangular'
  • no additional fields are required
• 'gaussian'
  • tFWHM - FWHM, in microseconds
  • trunc - truncation parameter (0 to 1) (alternative to tFWHM)
• 'sinc'
  • zerocross - width between the first zero-crossing points in microseconds
• 'sech'
  • beta - dimensionless truncation parameter (used as beta/tp)
  • n - exponent of the secant function argument (default = 1)

    asymmetric sech pulses can be obtained by specifying two values for n
    to get amplitude- and frequency-modulated higher-order sech pulses (HSn, n>1) with offset-independent adiabaticity, combine the sech AM function with the uniformQ FM option (see below)

• 'WURST'
  • nwurst - parameter determining the steepness of the amplitude function
• 'halfsin'
• no additional fields are required
• 'quartersin'
• trise - rise time in microseconds for quarter sine weighting at the pulse edges
• 'tanh2'
• trise - rise time in microseconds for tanh² weighting at the pulse edges
• 'GaussianCascade'

If Par.Type is set to 'G3', 'G4', 'Q3' or 'Q5', the Gaussian pulse cascades are computed with the parameters listed in Emsley, L., Bodenhausen, G., Gaussian pulse cascades: New analytical functions for rectangular selective inversion and in-phase excitation in NMR, Chem. Phys. Lett. 165, 469-476 (1990), DOI: 10.1016/0009-2614(90)87025-M (Table 1 on p. 473, Cycle 3 or Table 2 on p. 474, Cycle 11) and Emsley, L., Bodenhausen, G., Optimization of shaped selective pulses for NMR using a quaternion description of their overall propagators, J. Magn. Reson. 97, 135-148 (1992), DOI: 10.1016/0022-2364(92)90242-Y (Table 2 on p. 142), respectively. Otherwise the following parameters need to be specified:

• A0 - list of relative amplitudes
• x0 - list of positions (in fractions of the pulse length)
• FWHM - list of FWHM (in fractions of the pulse length)
• 'FourierSeries'

If Par.Type is set to 'I-BURP 1', 'I-BURP 2', 'E-BURP 1', 'E-BURP 2', 'U-BURP', 'RE-BURP', 'SNOB i2' or 'SNOB i3', the finite Fourier series pulses are computed with the coefficients listed in Geen, H., Freeman, R., Band-selective radiofrequency pulses, J. Magn. Reson. 93, 93-141 (1991), DOI: 10.1016/0022-2364(91)90034-Q(Table 3 for E-BURP 1, Table 4 for E-BURP 2, Table 5 for I-BURP 1, Table 6 for I-BURP 2, Table 7 for U-BURP and Table 8 for RE-BURP, N_p = 256) and Kupce, E., Boyd, J., Campbell, I. D., Short Selective Pulses for Biochemical Applications, J. Magn. Reson. B 106, 300-303 (1995), DOI: 10.1006/jmrb.1995.1049 (Table 1 on p. 300 for SNOB i2 and SNOB i3). Otherwise the following parameters need to be specified:

• A0 - initial amplitude coefficient
• An - list of Fourier coefficients for the cosine term
• Bn - list of Fourier coefficients for the sine term

For the definition of the FM functions, in all cases the frequency sweep range needs to be defined in Par.Frequency with start and end frequencies. Also, depending on the FM function, the following additional fields are required:

• 'none'
  • no additional fields are required
• 'linear'
  • no additional fields are required
• 'tanh'
  • beta - dimensionless truncation parameter (see 'sech')
• 'uniformQ'
  • no additional fields are required. This option calculates the FM function for an offset-independent adiabaticity (OIA) pulse given the specified AM function (see below for details).

I and Q data describing an arbitrary pulse shape not included in the predefined options listed below. If only one component is defined, the other is set to zero. The time axis is reconstructed based on Par.tp and the length of the I and Q or IQ vectors. All other input parameters (Par.TimeStep, Par.Flip, Par.Amplitude, Par.Qcrit, Par.Frequency, Par.Phase, Par.Type) are ignored.

With this input, pulse can be used to apply bandwidth compensation to user-defined pulse shapes.

Microwave frequency (LO), in GHz.

This gives the frequency response of the resonator, in the form Par.FrequencyResponse = [Frequency; TransferFunction] with the (possibly experimental) resonator transfer function in TransferFunction and the corresponding frequency axis in Frequency (in GHz). A complex transfer function input in Par.FrequencyResponse is used directly in the bandwidth compensation. A real transfer function input is assumed to correspond to the magnitude response, and the associated phase response is estimated (see details below).

If FrequencyResponse is given, the alternative input fields ResonatorFrequency and ResonatorQL are ignored.

Resonator center frequency in GHz and loaded Q-factor for the calculation of an ideal transfer function modeling the resonator as a RLC series circuit. Example:

Par.ResonatorFrequency = 34.9;  % resonator center frequency, GHz
Par.ResonatorQL = 100;     % loaded Q-factor of resonator

If FrequencyResponse is given, it takes precedence over ResonatorFrequency and ResonatorQL.

Oversampling factor for the automatic determination of the time step (default = 10).