Geometrical interpretation of lifetimes#
Demonstrate the geometrical interpretation of fluorescence lifetimes in the phasor plot.
The PhasorPy library is used to demonstrate the geometrical interpretation of fluorescence lifetimes and other quantities in the phasor plot:
single exponential lifetimes
fractional intensities of lifetime components
apparent single lifetime from phase and modulation
normal lifetime
phase and modulation
phasor coordinates
Import required modules and functions:
import numpy
from phasorpy.phasor import (
phasor_from_lifetime,
phasor_to_apparent_lifetime,
phasor_to_normal_lifetime,
phasor_to_polar,
)
from phasorpy.plot import PhasorPlot
Calculate phasor coordinates, polar coordinates, and apparent lifetimes from a set of lifetimes and their fractional intensities at a given frequency:
frequency = 80.0 # MHz
lifetime = (0.4, 4.0) # ns
fraction = (0.6, 0.4)
real, imag = phasor_from_lifetime(frequency, lifetime, fraction)
phase, modulation = phasor_to_polar(real, imag)
component_real, component_imag = phasor_from_lifetime(frequency, lifetime)
tau_phi, tau_mod = phasor_to_apparent_lifetime(real, imag, frequency)
tau_norm = phasor_to_normal_lifetime(real, imag, frequency)
tau_phi_re, tau_phi_im = phasor_from_lifetime(frequency, tau_phi)
tau_mod_re, tau_mod_im = phasor_from_lifetime(frequency, tau_mod)
tau_norm_re, tau_norm_im = phasor_from_lifetime(frequency, tau_norm)
Plot the phasor coordinates and lifetimes, and annotate the plot:
color_phasor = 'black'
color_component = 'tab:orange'
color_phase = 'tab:blue'
color_modulation = 'tab:red'
color_normal = 'tab:green'
fontsize = 18
linewidth = 3.0
textoffset = 0.01
plot = PhasorPlot(
title='Geometrical interpretation of lifetimes in the phasor plot',
frequency=frequency,
xlabel='',
ylabel='',
xticks=[0, 0.5, 1],
yticks=[0, 0.5],
xticklabels=['0', '1/2', '1'],
yticklabels=['0', '1/2'],
xlim=(0.0, 1.01),
ylim=(0.0, 0.6),
grid=False,
)
plot.ax.tick_params(axis='both', which='major', labelsize=fontsize * 2 / 3)
plot.ax.spines['top'].set_visible(False)
plot.ax.spines['right'].set_visible(False)
# plot.ax.spines['left'].set_linewidth(linewidth)
# plot.ax.spines['bottom'].set_linewidth(linewidth)
plot.semicircle(
# frequency,
linewidth=linewidth,
color='tab:gray',
zorder=0,
)
plot.line(
(real, real),
(0.0, imag),
linestyle='--',
color=color_phasor,
linewidth=1,
zorder=0,
)
plot.line(
(real, 0),
(imag, imag),
linestyle='--',
color=color_phasor,
linewidth=1,
zorder=0,
)
component_real = numpy.atleast_1d(component_real)
component_imag = numpy.atleast_1d(component_imag)
for i in range(len(lifetime)):
plot.arrow(
(real, imag),
(component_real[i], component_imag[i]),
color=color_component,
linewidth=linewidth,
)
plot.arrow(
(modulation / 4, 0.0),
(real / 4, imag / 4),
angle=phase,
color=color_phase,
arrowstyle='-',
linewidth=linewidth,
)
plot.arrow(
(0.5, 0.0),
(real, imag),
color=color_normal,
arrowstyle='-',
linewidth=linewidth,
)
plot.arrow(
(modulation, 0.0),
(real, imag),
angle=phase,
color=color_modulation,
arrowstyle='-',
linewidth=linewidth,
)
plot.arrow(
(real, imag),
(tau_phi_re, tau_phi_im),
color=color_phase,
linewidth=linewidth,
)
plot.arrow(
(real, imag),
(tau_mod_re, tau_mod_im),
angle=phase,
color=color_modulation,
linewidth=linewidth,
)
plot.arrow(
(real, imag),
(tau_norm_re, tau_norm_im),
color=color_normal,
linewidth=linewidth,
)
plot.arrow(
(0, 0),
(real, imag),
color=color_phase,
linewidth=linewidth,
arrowstyle='-',
# zorder=1,
)
plot.plot(
real,
imag,
marker='o',
markersize=12,
markeredgewidth=linewidth,
markeredgecolor=color_phasor,
markerfacecolor='white',
zorder=1,
)
plot.ax.text(
-textoffset,
imag,
'$S$',
fontsize=fontsize,
ha='right',
va='center',
color=color_phasor,
)
plot.ax.text(
real,
-textoffset,
'$G$',
fontsize=fontsize,
ha='center',
va='top',
color=color_phasor,
)
plot.ax.text(
modulation / 4 + textoffset,
imag / 8,
'$\\varphi$',
fontsize=fontsize,
ha='left',
va='center',
color=color_phase,
)
plot.ax.text(
modulation,
-textoffset,
'$M$',
fontsize=fontsize,
ha='center',
va='top',
color=color_modulation,
)
if len(lifetime) == 2:
for i in range(len(lifetime)):
plot.ax.text(
real + (component_real[i] - real) / 2,
imag + (component_imag[i] - imag) / 2 - textoffset,
'$\\alpha_{%i}$' % i,
fontsize=fontsize,
ha='center',
va='top',
color=color_component,
)
plot.components(
component_real,
component_imag,
fraction=fraction,
labels=['$\\tau_{%d}$' % i for i in range(len(lifetime))],
fontsize=fontsize,
ls=' ',
color=color_component,
)
plot.components(
tau_phi_re,
tau_phi_im,
labels=['$\\tau_{\\varphi}$'],
fontsize=fontsize,
color=color_phase,
)
plot.components(
tau_mod_re,
tau_mod_im,
labels=['$\\tau_{M}$'],
fontsize=fontsize,
color=color_modulation,
)
plot.components(
tau_norm_re,
tau_norm_im,
labels=['$\\tau_{N}$'],
fontsize=fontsize,
color=color_normal,
)
plot.show()
The figure demonstrates:
single exponential lifetimes \(\tau_{i}\) correspond to phasor coordinates on the universal semicircle.
the phasor coordinates \(G\) and \(S\) of a mixture of two components with single-exponential lifetimes, weighted by their fractional intensities \(\alpha_{i}\), lie on a line between the phasor coordinates of the single components.
the phase \(\varphi\) of the phasor coordinates \(G\) and \(S\) is the angle of the phasor coordinates with respect to the origin.
the modulation \(M\) of the phasor coordinates \(G\) and \(S\) is the distance from the origin to the phasor coordinates.
the apparent single lifetime from phase \(\tau_{\varphi}\) of the component mixture is the single exponential lifetime corresponding to the intersection of the universal circle with a line through the origin and the phasor coordinates \(G\) and \(S\).
the apparent single lifetime from modulation \(\tau_{M}\) of the component mixture is the single exponential lifetime corresponding to the intersection of the universal circle with a circle around the origin of radius equal to the modulation \(M\).
the normal lifetime \(\tau_{N}\) of the component mixture is the single exponential lifetime corresponding to the nearest point on the universal circle to the phasor coordinates \(G\) and \(S\), which is the intersection of the universal circle with the line through the center of the universal circle and the phasor coordinates \(G\) and \(S\).
sphinx_gallery_thumbnail_number = -1 mypy: allow-untyped-defs, allow-untyped-calls mypy: disable-error-code=”arg-type, assignment”
Total running time of the script: (0 minutes 0.136 seconds)