Theory

%matplotlib inline

import numpy as np
from scipy.special import jv as J
from scipy.integrate import cumulative_trapezoid
from ipywidgets import interact
import matplotlib.pyplot as plt


def plot_fields(E0=1.0, beta=1.0, m=0, k=2.0, N=5, R=0.0, Phi=0.1, d=5.0):
    Z = np.linspace(0, 10, 100)

    def Ez(z=Z, r=R, phi=Phi):
        return E0 * np.exp(-1j*beta*Z) * J(m, (k**2-beta**2)**0.5*R) * np.cos(m*Phi)

    mag_Ez = Ez()

    def En(n):
        Z = np.linspace(-d/2, d/2, 100)
        return cumulative_trapezoid(Ez(Z, 0, Phi*np.exp(1j*(beta+2*np.pi*n/d)*Z)))[-1] / d

    Ez_sum = sum([En(n) * np.exp(-1j*(beta+2*np.pi*n/d)*Z) * J(m, (k**2-(beta+2*np.pi*n/d)**2)**0.5*R) * np.cos(m*Phi) for n in range(-N, N+1, 1)])
    mag_Ez_sum = np.abs(Ez_sum)

    fig, axs = plt.subplots(1, 2, figsize=(12, 5))

    cax1 = axs[0].plot(Z, mag_Ez)
    axs[0].set_title("Magnitude of Ez")
    axs[0].set_xlabel("z")
    axs[0].set_ylabel("Ez")

    cax2 = axs[1].plot(Z, mag_Ez_sum)
    axs[1].set_title("Magnitude of Ez_sum")
    axs[1].set_xlabel("z")
    axs[1].set_ylabel("Ez_sum")

    plt.tight_layout()
    fig.show()


interact(plot_fields,
         E0=(0.1, 2.0, 0.1),
         beta=(0.1, 5.0, 0.1),
         m=(0, 10),
         k=(0.1, 5.0, 0.1),
         N=(0, 20),
         R=(0.0, 10.0, 0.1),
         Phi=(0.1, 10.0, 0.1),
         d=(1.0, 10.0, 0.1))

<function __main__.plot_fields(E0=1.0, beta=1.0, m=0, k=2.0, N=5, R=0.0, Phi=0.1, d=5.0)>