Calculates pulse shape and amplitude, frequency and phase modulation functions for a series of pulse types.
pulse(Par) pulse(Par,Opt) [t,IQ] = pulse(...) [t,IQ,modulation] = pulse(...)
This function calculates pulse shapes for predefined types of shaped pulses. The corresponding excitation profile can be computed with the function exciteprofile.
The output contains the time axis data in
t (time in microseconds).
IQ contains the pulse shape data with the in-phase component in the real part and the quadrature component in the imaginary part.
The third (optional) output argument contains the structure
modulation with the calculated modulation functions for the pulse:
modulation.A: amplitude modulation function (in MHz)
modulation.freq: frequency modulation function (in MHz)
modulation.phase: phase modulation function (in radians)
If you don't request any output,
pulse plots the calculated pulse shape, modulation functions and excitation profiles.
The input arguments to
Par, a structure containing pulse parameter definitions, and
Opt, an optional structure containing options for the calculation.
Par is a structure containing all parameters necessary to specify the pulse shape. The available fields are listed below.
Par.Flip = piis used as default.
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, 13570–13582 (2015) (DOI: 10.1021/acs.jpcb.5b02964) is used.
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, 13570–13582 (2015) (DOI: 10.1021/acs.jpcb.5b02964).
Par.Frequency = -10; % constant frequency offset Par.Frequency = [-50 50]; % up-sweep Par.Frequency = [50 20]; % down-sweep
pi/2for +y, etc. If not given, it defaults to 0.
'AM'), where AM refers to the amplitude modulation function and FM to the frequency modulation function.
The available amplitude modulation functions are
'FourierSeries'. Different AM functions can be multiplied by concatenating the keywords as
'AM1*AM2/FM'. The default is
The available options for frequency modulation are
'uniformQ'. If only an AM keyword is given, the FM function is set to
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.
Depending on the AM function, additional fields need to be specified:
tFWHM- FWHM, in microseconds
trunc- truncation parameter (0 to 1) (alternative to
zerocross- width between the first zero-crossing points in microseconds
beta- dimensionless truncation parameter (used as
n- exponent of the secant function argument (default = 1)
asymmetric sech pulses can be obtained by specifying two values for
nwurst- parameter determining the steepness of the amplitude function
trise- rise time in microseconds for quarter sine weighting at the pulse edges
Par.Type is set to
'Q3', 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) 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)
Par.Type is set to
'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 5 on p. 117 for I-BURP 1 and Table 6 on p. 119 for I-BURP 2, Np = 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:
beta- dimensionless truncation parameter (see
Par.tpand the length of the
IQvectors. All other input parameters (
Par.Type) are ignored.
With this input,
pulse can be used to apply bandwidth compensation to user-defined pulse shapes.
For frequency-swept pulses,
pulse can compensate for the resonator bandwidth to achieve offset-independent adiabaticity. In this case, the following additional parameters have to be included in the
Par.FrequencyResponse = [Frequency; TransferFunction]with the (possibly experimental) resonator transfer function in
TransferFunctionand the corresponding frequency axis in
Frequency(in GHz). A complex transfer function input in
Par.FrequencyResponseis 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).
FrequencyResponse is given, the alternative input fields
ResonatorQL are ignored.
Par.ResonatorFreq = 34.9; % resonator center frequency, GHz Par.ResonatorQL = 100; % loaded Q-factor of resonator
FrequencyResponse is given, it takes precendence over
The parameters in the
Opt structure define different settings for the calculation. The available fields are given below.
The pulse amplitude, frequency and phase modulation functions are calculated using the following equations. The origin of the time variable t is at the center of the pulse and runs from -tp/2 to +tp/2.
Amplitude modulation functions:
Frequency and phase modulation functions (BW is the difference between end and start frequencies):
The frequency modulation function is calculated as the integral of the squared amplitude modulation function and multiplied by the specified bandwidth (centered at zero). This can be used for nth order sech pulses or in general to obtain offset-independent adiabaticity pulses given a certain amplitude function (see Garwood, M., DelaBarre, L., J. Magn. Reson. 153, 155-177 (2001), DOI: 10.1006/jmre.2001.2340). The phase modulation function is obtained by integration and the phase is set to zero at the center of the pulse.
The time axis for the calculation of the modulation functions is centered at zero. The normalized amplitude modulation functions given above are multiplied by
The final pulse shape is calculated as:
where the different terms are the amplitude modulation, the phase modulation, the center frequency and the phase offset defined in
Compensation for the resonator bandwidth by adaptation of the chirp rates in frequency-swept pulses is implemented as described in Doll, A., Pribitzer, S., Tschaggelar, R., Jeschke, G., Adiabatic and fast passage ultra-wideband inversion in pulsed EPR, J. Magn. Reson. 230, 27-39 (2013), DOI: 10.1016/j.jmr.2013.01.002 and Pribitzer, S., Doll, A., Jeschke, G., SPIDYAN, a MATLAB library for simulating pulse EPR experiments with arbitrary waveform excitation. J. Magn. Reson. 263, 45-54 (2016) (DOI: 10.1016/j.jmr.2015.12.014). For pulses with uniform adiabaticity (sech/tanh and any pulse with a frequency modulation function set to
'uniformQ') both the frequency and the amplitude modulation functions defined above are adjusted to compensate for the resonator profile. For all other pulses, the amplitude modulation function is preserved and only the frequency modulation function is adapted to the provided resonator profile. The phase modulation function is obtained by integration and the phase is set to zero at the center of the pulse.
exciteprofile, rfmixer, transmitter, resonator, evolve, saffron