Dampled Oscillation

Objective

Demonstrates dmp_osc_conv_gau, dmp_osc_conv_cauchy, dmp_osc_conv_pvoigt routine

dmp_osc_conv_gau: Computes the convolution of damped oscillation and gaussian function

dmp_osc_conv_cauchy: Computes the convolution of damped oscillation and cauchy function

dmp_osc_conv_pvoigt: Computes the convolution of damped oscillation and pseudo voigt function

Dampled oscillation is modeled to \(\exp(-kt)\cos(2\pi t/T+\phi)\)

import numpy as np
import matplotlib.pyplot as plt
from TRXASprefitpack import dmp_osc_conv_gau, dmp_osc_conv_cauchy, dmp_osc_conv_pvoigt
plt.rcParams["figure.figsize"] = (14,10)

Condition

  1. fwhm: 0.15 ps

  2. tau: 10 ps

  3. period: 0.15 ps 0.3 ps 3 ps

  4. phase: 0

  5. eta: 0.3

fwhm = 0.15
tau = 10
period = [0.15, 0.3, 3]
phase_factor = 0
eta = 0.3

# time range
t_0 = np.arange(-2, -1, 0.2)
t_1 = np.arange(-1, 1, 0.1)
t_2 = np.arange(1, 3, 0.2)
t_3 = np.arange(3, 10, 0.5)
t_4 = np.arange(10, 50, 4)
t_5 = np.arange(50, 100, 10)
t_6 = np.arange(100, 1100, 100)

t = np.hstack((t_0, t_1, t_2, t_3, t_4, t_5, t_6))

dmp_osc_conv_gau routine

help(dmp_osc_conv_gau)

Help on function dmp_osc_conv_gau in module TRXASprefitpack.mathfun.exp_conv_irf:

dmp_osc_conv_gau(t: Union[float, numpy.ndarray], fwhm: float, k: float, T: float, phase: float) -> Union[float, numpy.ndarray]
    Compute damped oscillation convolved with normalized gaussian
    distribution
    
    Args:
      t: time
      fwhm: full width at half maximum of gaussian distribution
      k: damping constant (inverse of life time)
      T: period of vibration 
      phase: phase factor
    
    Returns:
     Convolution of normalized gaussian distribution and 
     damped oscillation :math:`(\exp(-kt)cos(2\pi t/T+phase))`.

gau_osc_1 = dmp_osc_conv_gau(t, fwhm, 1/tau, period[0], phase_factor)
gau_osc_2 = dmp_osc_conv_gau(t, fwhm, 1/tau, period[1], phase_factor)
gau_osc_3 = dmp_osc_conv_gau(t, fwhm, 1/tau, period[2], phase_factor)

plt.plot(t, gau_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, gau_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, gau_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-1, 2)
plt.show()

png

plt.plot(t, gau_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, gau_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, gau_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-1,50)
plt.show()

png

plt.plot(t, gau_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, gau_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, gau_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-2, 1000)
plt.show()

png

dmp_osc_conv_cauchy routine

help(dmp_osc_conv_cauchy)

Help on function dmp_osc_conv_cauchy in module TRXASprefitpack.mathfun.exp_conv_irf:

dmp_osc_conv_cauchy(t: Union[float, numpy.ndarray], fwhm: float, k: float, T: float, phase: float) -> Union[float, numpy.ndarray]
    Compute damped oscillation convolved with normalized cauchy
    distribution
    
    Args:
      t: time
      fwhm: full width at half maximum of cauchy distribution
      k: damping constant (inverse of life time)
      T: period of vibration 
      phase: phase factor
    
    Returns:
     Convolution of normalized cauchy distribution and 
     damped oscillation :math:`(\exp(-kt)cos(2\pi t/T+phase))`.

cauchy_osc_1 = dmp_osc_conv_cauchy(t, fwhm, 1/tau, period[0], phase_factor)
cauchy_osc_2 = dmp_osc_conv_cauchy(t, fwhm, 1/tau, period[1], phase_factor)
cauchy_osc_3 = dmp_osc_conv_cauchy(t, fwhm, 1/tau, period[2], phase_factor)

plt.plot(t, cauchy_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, cauchy_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, cauchy_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-1, 2)
plt.show()

png

plt.plot(t, cauchy_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, cauchy_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, cauchy_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-1, 50)
plt.show()

png

plt.plot(t, cauchy_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, cauchy_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, cauchy_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-2, 1000)
plt.show()

png

dmp_osc_conv_pvoigt

help(dmp_osc_conv_pvoigt)

Help on function dmp_osc_conv_pvoigt in module TRXASprefitpack.mathfun.exp_conv_irf:

dmp_osc_conv_pvoigt(t: Union[float, numpy.ndarray], fwhm_G: float, fwhm_L: float, eta: float, k: float, T: float, phase: float) -> Union[float, numpy.ndarray]
    Compute damped oscillation convolved with normalized pseudo
    voigt profile (i.e. linear combination of normalized gaussian and
    cauchy distribution)
    
    :math:`\eta C(\mathrm{fwhm}_L, t) + (1-\eta)G(\mathrm{fwhm}_G, t)`
    
    Args:
       t: time
       fwhm_G: full width at half maximum of gaussian part of
               pseudo voigt profile
       fwhm_L: full width at half maximum of cauchy part of
               pseudo voigt profile
       eta: mixing parameter
       k: damping constant (inverse of life time)
       T: period of vibration 
       phase: phase factor
    
    Returns:
     Convolution of normalized pseudo voigt profile and
     damped oscillation :math:`(\exp(-kt)cos(2\pi t/T+phase))`.

pvoigt_osc_1 = dmp_osc_conv_pvoigt(t, fwhm, fwhm, eta, 1/tau, period[0], phase_factor)
pvoigt_osc_2 = dmp_osc_conv_pvoigt(t, fwhm, fwhm, eta, 1/tau, period[1], phase_factor)
pvoigt_osc_3 = dmp_osc_conv_pvoigt(t, fwhm, fwhm, eta, 1/tau, period[2], phase_factor)

plt.plot(t, pvoigt_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, pvoigt_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, pvoigt_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-1, 2)
plt.show()

png

plt.plot(t, cauchy_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, cauchy_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, cauchy_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-1, 50)
plt.show()

png

plt.plot(t, cauchy_osc_1, label=f'period: {period[0]} ps')
plt.plot(t, cauchy_osc_2, label=f'period: {period[1]} ps')
plt.plot(t, cauchy_osc_3, label=f'period: {period[2]} ps')
plt.legend()
plt.xlim(-2, 1000)
plt.show()

png

Conclusion

  1. When the value of period of oscillation is similar to or less than fwhm of gaussian probe pulse, It is hard to see oscillation feature.

  2. Shown as IRF example, the shape of the convolution of cauchy irf and damped oscilliation is much broader than that of gaussian irf one.

  3. Eventhough phase is 0, the oscillation peak near 0 is not the highest one when oscillation period T is comparable to fwhm