Source code for mbtrack2.utilities.spectrum

# -*- coding: utf-8 -*-
"""
Module where bunch and beam spectrums and profile are defined.
"""

import numpy as np
from scipy.special import jv, spherical_jn


[docs]def spectral_density(frequency, sigma, m=1, k=0, mode="Hermite"): """ Compute the spectral density of different modes for various values of the head-tail mode number, based on Table 1 p238 of [1]. Parameters ---------- frequency : list or numpy array sample points of the spectral density in [Hz] sigma : float RMS bunch length in [s] m : int, optional head-tail (or azimutal/synchrotron) mode number k : int, optional radial mode number (such that |q|=m+2k, where |q| is the head-tail mode number) mode: str, optional type of the mode taken into account for the computation: -"Hermite" modes for Gaussian bunches (typical for electrons) -"Chebyshev" for airbag bunches -"Legendre" for parabolic bunches (typical for protons) -"Sacherer" or "Sinusoidal" simplifying approximation of Legendre modes from [3] Returns ------- numpy array References ---------- [1] : Handbook of accelerator physics and engineering, 3rd printing. [2] : Ng, K. Y. (2005). Physics of intensity dependent beam instabilities. WORLD SCIENTIFIC. https://doi.org/10.1142/5835 [3] : Sacherer, F. J. (1972). Methods for computing bunched beam instabilities. CERN Tech. rep. CERN/SI-BR/72-5 https://cds.cern.ch/record/2291670?ln=en """ if mode == "Hermite": return 1 / (np.math.factorial(m) * 2**m) * (2 * np.pi * frequency * sigma)**( 2 * m) * np.exp(-(2 * np.pi * frequency * sigma)**2) elif mode == "Chebyshev": tau_l = 4 * sigma return (jv(m, 2 * np.pi * frequency * tau_l))**2 elif mode == "Legendre": tau_l = 4 * sigma return (spherical_jn(m, np.abs(2 * np.pi * frequency * tau_l)))**2 elif mode == "Sacherer" or mode == "Sinusoidal": y = 4 * 2 * np.pi * frequency * sigma / np.pi return (2 * (m+1) / np.pi * 1 / np.abs(y**2 - (m + 1)**2) * np.sqrt(1 + (-1)**m * np.cos(np.pi * y)))**2 else: raise NotImplementedError("Not implemanted yet.")
[docs]def gaussian_bunch_spectrum(frequency, sigma): """ Compute a Gaussian bunch spectrum [1]. Parameters ---------- frequency : array sample points of the beam spectrum in [Hz]. sigma : float RMS bunch length in [s]. Returns ------- bunch_spectrum : array Bunch spectrum sampled at points given in frequency. References ---------- [1] : Gamelin, A. (2018). Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX. p86, Eq. 4.19 """ return np.exp(-1 / 2 * (2 * np.pi * frequency)**2 * sigma**2)
[docs]def gaussian_bunch(time, sigma): """ Compute a Gaussian bunch profile. Parameters ---------- time : array sample points of the bunch profile in [s]. sigma : float RMS bunch length in [s]. Returns ------- bunch_profile : array Bunch profile in [s**-1] sampled at points given in time. """ return np.exp(-1 / 2 * (time**2 / sigma**2)) / (sigma * np.sqrt(2 * np.pi))
[docs]def beam_spectrum(frequency, M, bunch_spacing, sigma=None, bunch_spectrum=None): """ Compute the beam spectrum assuming constant spacing between bunches [1]. Parameters ---------- frequency : list or numpy array sample points of the beam spectrum in [Hz]. M : int Number of bunches. bunch_spacing : float Time between two bunches in [s]. sigma : float, optional If bunch_spectrum is None then a Gaussian bunch with sigma RMS bunch length in [s] is assumed. bunch_spectrum : array, optional Bunch spectrum sampled at points given in frequency. Returns ------- beam_spectrum : array References ---------- [1] Rumolo, G - Beam Instabilities - CAS - CERN Accelerator School: Advanced Accelerator Physics Course - 2014, Eq. 9 """ if bunch_spectrum is None: bunch_spectrum = gaussian_bunch_spectrum(frequency, sigma) beam_spectrum = (bunch_spectrum * np.exp(1j * np.pi * frequency * bunch_spacing * (M-1)) * np.sin(M * np.pi * frequency * bunch_spacing) / np.sin(np.pi * frequency * bunch_spacing)) return beam_spectrum