Appendices

Below is a set of python codes associated with the Appendices of Daniele Pelliccia and David M. Paganin, “Synchrotron Light: A Physics Journey from Laboratory to Cosmos” (Oxford University Press, 2025).

In order to run any of these python codes, you will need to include the following header.

from scipy import special
import numpy as np
import matplotlib.pyplot as plt
import math

synchrotron_style = {
    'font.family': 'FreeSerif',
    'font.size': 30,
    'axes.linewidth': 2.0,
    'axes.edgecolor': 'black',
    'grid.color': '#5b5b5b',
    'grid.linewidth': 1.2,
}

from scipy import special

import numpy as np

import matplotlib.pyplot as plt

import math

synchrotron_style = {

'font.family': 'FreeSerif',

'font.size': 30,

'axes.linewidth': 2.0,

'axes.edgecolor': 'black',

'grid.color': '#5b5b5b',

'grid.linewidth': 1.2,

}

Appendix A: transformation linking the angular variables $\varphi$ and

$\psi$

See Fig. A.1

phi = np.linspace(-np.pi, np.pi, 361)

with plt.style.context(('seaborn-v0_8-whitegrid')):
    plt.rcParams.update(synchrotron_style)
    fig = plt.figure(figsize=(8.5,6.5))
    for eps in [0, 0.5, 0.999]:
        plt.plot(phi, phi-eps*np.sin(phi), lw=3)   
    
    plt.xticks([-np.pi, -np.pi/2, 0, np.pi/2, np.pi], ['-$\\pi$', '-$\\pi/2$', '0', '$\\pi/2$', '$\\pi$'], fontsize = 26)
    plt.yticks([-np.pi, -np.pi/2, 0, np.pi/2, np.pi], ['-$\\pi$', '-$\\pi/2$', '0', '$\\pi/2$', '$\\pi$'], fontsize = 26)
    plt.grid(True)

    plt.xlabel('$\\varphi$')
    plt.ylabel('$\\psi$')   
    plt.show()

phi = np.linspace(-np.pi, np.pi, 361)

with plt.style.context(('seaborn-v0_8-whitegrid')):

plt.rcParams.update(synchrotron_style)

fig = plt.figure(figsize=(8.5,6.5))

for eps in [0, 0.5, 0.999]:

plt.plot(phi, phi-eps*np.sin(phi), lw=3)

plt.xticks([-np.pi, -np.pi/2, 0, np.pi/2, np.pi], ['-$\\pi$', '-$\\pi/2$', '0', '$\\pi/2$', '$\\pi$'], fontsize = 26)

plt.yticks([-np.pi, -np.pi/2, 0, np.pi/2, np.pi], ['-$\\pi$', '-$\\pi/2$', '0', '$\\pi/2$', '$\\pi$'], fontsize = 26)

plt.grid(True)

plt.xlabel('$\\varphi$')

plt.ylabel('$\\psi$')

plt.show()

Appendix B: Boundary conditions for perfectly reflecting cavity walls

See Fig. B.1.

ell = 1
m = 1
x1 = np.linspace(-0.2, 0, 20)
x0 = np.linspace(0, ell, 100)
x2 = np.linspace(ell, 1.2, 20)

with plt.style.context(('seaborn-v0_8-whitegrid')):
    plt.rcParams.update(synchrotron_style)
    fig, ax = plt.subplots(1, 1,figsize=(10,8))
    plt.plot(x0, np.sin(np.pi*m*x0/1), 'b', lw=4, label = "$m=10$")
    plt.plot(x0, np.sin(2*np.pi*m*x0/1), 'r', lw=4, label = "$m=20$")
    plt.plot(x2, -np.sin(np.pi*m*x2/1), 'b--', lw=4)
    plt.plot(x1, -np.sin(np.pi*m*x1/1), 'b--', lw=4)
    plt.plot(x2, np.sin(2*np.pi*m*x2/1), 'r--', lw=4)
    plt.plot(x1, np.sin(2*np.pi*m*x1/1), 'r--', lw=4)
    
    ax.set_xticks([0,0.5,1])
    ax.set_xticklabels(["0", "0.5", "1"], fontsize=30)
    ax.set_yticks([-1,0, 1])
    ax.set_yticklabels(["-1", "0","1"], fontsize=30)        
    plt.xlabel('$x$', fontsize=20)
    plt.ylabel('$\\sin(\\pi mx/\\ell)$', fontsize=26)
    plt.legend(fontsize=26, handlelength=1, frameon=True, framealpha=1, \
               loc = 'center left', bbox_to_anchor=(0.17, 0.2))
    plt.show()

ell = 1

m = 1

x1 = np.linspace(-0.2, 0, 20)

x0 = np.linspace(0, ell, 100)

x2 = np.linspace(ell, 1.2, 20)

with plt.style.context(('seaborn-v0_8-whitegrid')):

plt.rcParams.update(synchrotron_style)

fig, ax = plt.subplots(1, 1,figsize=(10,8))

plt.plot(x0, np.sin(np.pi*m*x0/1), 'b', lw=4, label = "$m=10$")

plt.plot(x0, np.sin(2*np.pi*m*x0/1), 'r', lw=4, label = "$m=20$")

plt.plot(x2, -np.sin(np.pi*m*x2/1), 'b--', lw=4)

plt.plot(x1, -np.sin(np.pi*m*x1/1), 'b--', lw=4)

plt.plot(x2, np.sin(2*np.pi*m*x2/1), 'r--', lw=4)

plt.plot(x1, np.sin(2*np.pi*m*x1/1), 'r--', lw=4)

ax.set_xticks([0,0.5,1])

ax.set_xticklabels(["0", "0.5", "1"], fontsize=30)

ax.set_yticks([-1,0, 1])

ax.set_yticklabels(["-1", "0","1"], fontsize=30)

plt.xlabel('$x$', fontsize=20)

plt.ylabel('$\\sin(\\pi mx/\\ell)$', fontsize=26)

plt.legend(fontsize=26, handlelength=1, frameon=True, framealpha=1, \

loc = 'center left', bbox_to_anchor=(0.17, 0.2))

plt.show()

Appendix D: Relation between Taylor and Padé coefficients of $g(\tau)$

Calculation leading to the plot in Fig. D.1. See also Fig. 4.23.

def cn_recursive(n):
    # Base case: c_0 = 0
    if n == 0:
        return 0
    # Recursive case: c_n = c_{n-1}/2 + 1/2^(n-1)
    else:
        return 0.5*cn_recursive(n-1) + 1/2**(n-1)

def derivative_K(z, N):
    if N == 0:
        return z*special.kv(2/3,z/2)
    else:
        return cn_recursive(N)*special.kvp(2/3,z/2,n=N-1) + (z/2**N)*special.kvp(2/3,z/2,n=N)

def cn_recursive(n):

# Base case: c_0 = 0

if n == 0:

return 0

# Recursive case: c_n = c_{n-1}/2 + 1/2^(n-1)

else:

return 0.5*cn_recursive(n-1) + 1/2**(n-1)

def derivative_K(z, N):

if N == 0:

return z*special.kv(2/3,z/2)

else:

return cn_recursive(N)*special.kvp(2/3,z/2,n=N-1) + (z/2**N)*special.kvp(2/3,z/2,n=N)

x = np.linspace(0.25,10.25,401)

# Point of maximum
point_max = np.argmax(x*special.kv(2/3,x/2))

# Calculate the Taylor series up the the order ord
ord = 4
a = np.zeros(ord)
tayl_exp = np.zeros_like(x)
for i in range(ord):
    test = derivative_K(x, i)
    a[i] = test[30]/math.factorial(i)
    tayl_exp += a[i]*(x-x[30])**i

A = a[0]
D = (a[2]**2 - a[3]*a[1]) / (a[1]**2 -a[2]*a[0])
C = -a[0]*D/a[1] - a[2]/a[1]
B = a[1]+a[0]*C
xmax = 1.0+(np.sqrt(A**2*D**2 + B*D*(B-A*C)) - A*D)/(B*D)
pade = (A+B*(x-1)) / (1+C*(x-1)+D*(x-1)**2)

with plt.style.context(('seaborn-v0_8-whitegrid')):
    plt.rcParams.update(synchrotron_style)
    fig, axs = plt.subplots(1,2, figsize=(21,8))
    axs[0].plot(x, x*special.kv(2/3,x/2), 'k', lw=3,label = "Exact")
    axs[0].plot(x, tayl_exp, 'r', lw=3,label="Taylor")
    axs[0].plot(x, pade, 'b', lw=3,label="Pade'")
    axs[0].set_xlim([0.18, 1.8])
    axs[0].set_ylim([1, 1.25])
    
    axs[0].tick_params(axis='x', labelsize=36)
    axs[0].tick_params(axis='y', labelsize=36)
    axs[0].set_xlabel('$\\tau$', fontsize=36)
    axs[0].set_ylabel("$g(\\tau)$", fontsize=36)  
    axs[0].legend(fontsize = 34, frameon=True, framealpha=0.9, loc = 'upper right', labelspacing=0.3, handlelength=1)
    
    axs[1].plot(x, x*special.kv(2/3,x/2), 'k', lw=3,label = "Exact")
    axs[1].plot(x, tayl_exp, 'r', lw=3,label="Taylor")
    axs[1].plot(x, pade, 'b', lw=3,label="Pade'")
    axs[1].set_xlim([0.18, 10])
    axs[1].set_ylim([0.05, 1.25])
    
    axs[1].tick_params(axis='x', labelsize=32)
    axs[1].tick_params(axis='y', labelsize=32)
    axs[1].set_xlabel('$\\tau$', fontsize=36)
    axs[1].set_ylabel("$g(\\tau)$", fontsize=36)  
    axs[1].legend(fontsize = 34, frameon=True, framealpha=0.9, labelspacing=0.3, handlelength=1)

plt.show()

x = np.linspace(0.25,10.25,401)

# Point of maximum

point_max = np.argmax(x*special.kv(2/3,x/2))

# Calculate the Taylor series up the the order ord

ord = 4

a = np.zeros(ord)

tayl_exp = np.zeros_like(x)

for i in range(ord):

test = derivative_K(x, i)

a[i] = test[30]/math.factorial(i)

tayl_exp += a[i]*(x-x[30])**i

A = a[0]

D = (a[2]**2 - a[3]*a[1]) / (a[1]**2 -a[2]*a[0])

C = -a[0]*D/a[1] - a[2]/a[1]

B = a[1]+a[0]*C

xmax = 1.0+(np.sqrt(A**2*D**2 + B*D*(B-A*C)) - A*D)/(B*D)

pade = (A+B*(x-1)) / (1+C*(x-1)+D*(x-1)**2)

with plt.style.context(('seaborn-v0_8-whitegrid')):

plt.rcParams.update(synchrotron_style)

fig, axs = plt.subplots(1,2, figsize=(21,8))

axs[0].plot(x, x*special.kv(2/3,x/2), 'k', lw=3,label = "Exact")

axs[0].plot(x, tayl_exp, 'r', lw=3,label="Taylor")

axs[0].plot(x, pade, 'b', lw=3,label="Pade'")

axs[0].set_xlim([0.18, 1.8])

axs[0].set_ylim([1, 1.25])

axs[0].tick_params(axis='x', labelsize=36)

axs[0].tick_params(axis='y', labelsize=36)

axs[0].set_xlabel('$\\tau$', fontsize=36)

axs[0].set_ylabel("$g(\\tau)$", fontsize=36)

axs[0].legend(fontsize = 34, frameon=True, framealpha=0.9, loc = 'upper right', labelspacing=0.3, handlelength=1)

axs[1].plot(x, x*special.kv(2/3,x/2), 'k', lw=3,label = "Exact")

axs[1].plot(x, tayl_exp, 'r', lw=3,label="Taylor")

axs[1].plot(x, pade, 'b', lw=3,label="Pade'")

axs[1].set_xlim([0.18, 10])

axs[1].set_ylim([0.05, 1.25])

axs[1].tick_params(axis='x', labelsize=32)

axs[1].tick_params(axis='y', labelsize=32)

axs[1].set_xlabel('$\\tau$', fontsize=36)

axs[1].set_ylabel("$g(\\tau)$", fontsize=36)

axs[1].legend(fontsize = 34, frameon=True, framealpha=0.9, labelspacing=0.3, handlelength=1)

plt.show()

Appendix A: transformation linking the angular variables \varphi and

\psi

Appendix B: Boundary conditions for perfectly reflecting cavity walls

Appendix D: Relation between Taylor and Padé coefficients of g(\tau)

Appendix A: transformation linking the angular variables $\varphi$ and

$\psi$

Appendix D: Relation between Taylor and Padé coefficients of $g(\tau)$