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Shaped pulse in amplitude-frequency coordinates using Fokker-Planck formalism (Eqn. 33 in The pulse is assumed to be piecewise-constant and should be supplied with sufficiently fine time discretization to properly reproduce the waveform.




       L0          - drift Liouvillian that continues
                     running in the background

       Lx          - X projection of the RF operator

       Ly          - Y projection of the RF operator

       rho         - initial state vector or a stack 

       rf_frq_list - a vector of RF frequencies at each
                     time slice, Hz

       rf_amp_list - a vector of RF amplitudes at each
                     time slice, rad/s

       rf_dur_list - a vector of time slice durations,
                     in seconds

       rf_phi      - RF phase of the first pulse slice

       max_rank    - maximum rank of the Fokker-Planck
                     theory, increase until the answer 
                     stops changing, 2 is a good start

       method      - propagation method, 'expv' for Krylov
                     propagation, 'expm' for exponential
                     propagation, 'evolution' for Spinach
                     evolution function


       rho         - final state vector

       P           - effective pulse propagator, only 
                     available for the 'expm' method


An example of a chirped inversion pulse pulse applied to a system with 31 J-coupled protons (examples/nmr_liquids/shaped_pulse_3.m):

Chirp inversion.png

Note that only 100 time slices are required in the frequency-amplitude representation: considerably fewer than would be needed in the Cartesian representation used by shaped_pulse_xy.m function.


Of the three propagation methods, 'expv' is recommended because it runs Krylov propagation that avoids explicit matrix exponentiation. The 'expm' option forces the rather inefficient sparse matrix exponentiation path and should only be used when the effective propagator is required. In very anomalous cases (long pulses, large state vector stacks, very large state spaces), the 'evolution' option might become necessary.

See also

shaped_pulse_xy.m, read_wave.m, vg_pulse.m, pulse_shape.m, chirp_pulse_af.m, chirp_pulse_xy.m

Version 2.2, authors: Ilya Kuprov