Bunching¶

Bunching at some wavelength $\lambda$ for a list of particles $z$ is given by the weighted sum of complex phasors:

$$B(z, \lambda) \equiv \frac{\sum_j w_j e^{i k z_j}}{\sum w_j} $$

where $k = 2\pi/\lambda$ and $w_j$ are the weights of the particles.

See for example D. Ratner's disseratation.

Add bunching to particles¶

This uses a simple method to add perfect bunching at 0.1 µm

In [1]:

from pmd_beamphysics import ParticleGroup

%config InlineBackend.figure_format = 'retina'

from pmd_beamphysics import ParticleGroup %config InlineBackend.figure_format = 'retina'

In [2]:

P = ParticleGroup("data/bmad_particles2.h5")
P.drift_to_t()

wavelength = 0.1e-6
dz = (P.z / wavelength % 1) * wavelength
P.z -= dz

P = ParticleGroup("data/bmad_particles2.h5") P.drift_to_t() wavelength = 0.1e-6 dz = (P.z / wavelength % 1) * wavelength P.z -= dz

In [3]:

P.plot("z")

P.plot("z")

Calculate bunching¶

All of these methods will calculate the bunching. The first returns a complex number bunching

The string attributes return real numbers, magnitude and argument (phase):

• 'bunching_ returns np.abs(bunching)
• 'bunching_phase_ returns np.angle(bunching)

In [4]:

b = P.bunching(wavelength)
b

b = P.bunching(wavelength) b

Out[4]:

np.complex128(1-3.2133096564432656e-16j)

In [5]:

P["bunching_0.1e-6"]

P["bunching_0.1e-6"]

Out[5]:

np.float64(1.0)

In [6]:

P["bunching_phase_0.1e-6"]

P["bunching_phase_0.1e-6"]

Out[6]:

np.float64(-3.2133096564432656e-16)

In [7]:

P["bunching_0.1_um"]

P["bunching_0.1_um"]

Out[7]:

np.float64(1.0)

Simple plot¶

In [8]:

import numpy as np

import matplotlib.pyplot as plt

import numpy as np import matplotlib.pyplot as plt

In [9]:

wavelengths = wavelength * np.linspace(0.9, 1.1, 200)

wavelengths = wavelength * np.linspace(0.9, 1.1, 200)

In [10]:

plt.plot(wavelengths * 1e6, np.abs(list(map(P.bunching, wavelengths))))
plt.xlabel("wavelength (µm)")
plt.ylabel("bunching")

plt.plot(wavelengths * 1e6, np.abs(list(map(P.bunching, wavelengths)))) plt.xlabel("wavelength (µm)") plt.ylabel("bunching")

Out[10]:

Text(0, 0.5, 'bunching')

In [11]:

P.slice_plot("bunching_0.1_um")

P.slice_plot("bunching_0.1_um")

In [12]:

P.slice_plot("bunching_phase_0.1_um")

P.slice_plot("bunching_phase_0.1_um")

In [13]:

P.in_z_coordinates

P.in_z_coordinates

Out[13]:

False

Units¶

Bunching is dimensionless

In [14]:

P.units("bunching_0.1_um")

P.units("bunching_0.1_um")

Out[14]:

pmd_unit('', 1, (0, 0, 0, 0, 0, 0, 0))

In [15]:

P.units("bunching_phase_0.1_um")

P.units("bunching_phase_0.1_um")

Out[15]:

pmd_unit('rad', 1, (0, 0, 0, 0, 0, 0, 0))

Bunching function¶

This is the function that is used.

In [ ]:

In [16]:

?bunching

?bunching

Object `bunching` not found.