{"id":808,"date":"2025-07-20T04:07:09","date_gmt":"2025-07-20T04:07:09","guid":{"rendered":"https:\/\/synchrotron-light.net\/wp\/?page_id=808"},"modified":"2025-07-20T05:12:24","modified_gmt":"2025-07-20T05:12:24","slug":"chapter-11-calculation-of-synchrotron-radiation-from-first-principles","status":"publish","type":"page","link":"https:\/\/synchrotron-light.net\/wp\/?page_id=808","title":{"rendered":"Chapter 11: Calculation of synchrotron radiation from first principles"},"content":{"rendered":"

Below is a set of python codes associated with Chapter 11 of Daniele Pelliccia and David M. Paganin, “Synchrotron Light: A Physics Journey from Laboratory to Cosmos” (Oxford University Press, 2025).<\/p>\n

In order to run any of these python codes, you will need to include the following header.<\/p>\n

from scipy import special\r\nimport numpy as np\r\nimport matplotlib.pyplot as plt\r\n\r\nsynchrotron_style = {\r\n    'font.family': 'FreeSerif',\r\n    'font.size': 30,\r\n    'axes.linewidth': 2.0,\r\n    'axes.edgecolor': 'black',\r\n    'grid.color': '#5b5b5b',\r\n    'grid.linewidth': 1.2,\r\n}<\/pre>\n
Doppler shift<\/h2>\n
See Fig. 11.2<\/p>\n
c = 1. # set the speed of light in natural unit\r\nv1 = 0.9\r\nv2 = 0.999\r\nbeta1 = v1\/c\r\nbeta2 = v2\/c\r\ngamma1 = 1.0\/ np.sqrt(1.0 - v1**2\/c**2)\r\ngamma2 = 1.0\/ np.sqrt(1.0 - v2**2\/c**2)\r\n\r\ntheta = np.linspace(-np.pi\/2, np.pi\/2, 500, endpoint=False)\r\n\r\nfig = plt.figure(figsize=(9,6))\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    plt.plot(theta,1.0\/(gamma1*(1-beta1*np.cos(theta))), lw=4, label=\"$v = 0.9c$\")\r\n    plt.plot(theta,1.0\/(gamma2*(1-beta2*np.cos(theta))), lw=4, label=\"$v = 0.999c$\")\r\n    \r\n    plt.xlabel('$\\\\theta$ (rad)', fontsize=30, labelpad=0)\r\n    plt.ylabel(\"$\\\\frac{\\omega}{\\omega'}$\", rotation=0, fontsize=36, labelpad=25)\r\n    \r\n    plt.xticks(fontsize = 30)\r\n    plt.yticks(fontsize = 30)\r\n    plt.legend(fontsize = 28, frameon=True, handlelength=1, framealpha=1)<\/pre>\n
\"\"<\/p>\n
Dirac comb<\/h2>\n
This code snippet produces a basic version of the Dirac comb plotted in Fig. 11.3(b).<\/p>\n
N, n = 101, 30\r\ndef f(i):\r\n    return ((i-15) % n == 0) * 1\r\ncomb = np.fromfunction(f, (N,), dtype=int)\r\n\r\nplt.figure(figsize = (12,7))\r\nplt.plot(comb, lw=5)\r\nplt.axis('off')\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Spatial and temporal structure of synchrotron light<\/h2>\n
Archimedean spirals<\/h3>\n
See Fig. 11.11.<\/p>\n
phi = np.linspace(0,6*np.pi, 10000)\r\nr = [phi, phi, phi ]\r\noffset = [0, 2*np.pi\/3, 4*np.pi\/3]\r\nplt.figure(figsize=[20,20])\r\nfor i,radius in enumerate(r):\r\n    x = -radius*np.cos(phi-offset[i])\r\n    y = radius*np.sin(phi-offset[i])\r\n\r\n    plt.plot(x,y, lw=6)\r\n\r\nplt.axis(\"off\")    \r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Auxiliary function  f(s, \\beta) <\/span><\/h3>\n
See Figs. 11.12(a) and 11.13(a).<\/p>\n
bbeta = 0.8 # replace with 0.4 for Fig. 11.13(a)\r\nn = 1\r\ns = np.linspace(-np.pi,np.pi, 500)\r\naux = np.zeros_like(s)\r\nfor n in np.arange(-100,100,1):\r\n    aux = aux + n*np.exp(1j*n*s)*special.jvp(n, n*bbeta)\r\n    \r\nauxil = np.abs(aux)**2    \r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig,ax = plt.subplots(figsize=(10,8))\r\n    plt.plot(s,auxil, lw=4)\r\n    plt.xticks(fontsize = 24)\r\n    plt.yticks(fontsize = 26)\r\n    ax.set_xticks([-np.pi,-np.pi\/2, 0,np.pi\/2,np.pi],labels=[\"\", \"\", \"\", \"\",\"\"])\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Spatial structure of instantaneous energy density for a fixed instant of time<\/h3>\n
See Figs. 11.12(b) and 11.13(b).<\/p>\n
import tqdm\r\nbbeta = 0.8 # replace with 0.4 to generate Fig. 11.13(b)\r\nphi = np.linspace(0, 2*np.pi, 3600)\r\nr = np.arange(0, 20, 0.002)\r\n\r\nrr, theta = np.meshgrid(r, phi)\r\nvalues = np.random.random((phi.size, r.size))\r\n\r\ns = theta + rr\r\naux = np.zeros_like(s)\r\nn_max = 200\r\nfor n in tqdm.tqdm(np.arange(-n_max,n_max,1)):\r\n    aux = aux + n*np.exp(1j*n*s)*special.jvp(n, n*bbeta)\r\n\r\nauxil = np.abs(aux)**2   \r\n\r\nwith plt.style.context(('default')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    \r\n    fig, ax = plt.subplots(subplot_kw=dict(projection='polar'), figsize=(10,10))\r\n    ax.contourf(theta, rr, auxil, levels=200)\r\n    ax.set_rticks([0,5, 10,15,20])\r\n    ax.set_yticklabels([])\r\n    ax.tick_params(axis='y', labelsize=26)\r\n    ax.tick_params(axis='x', labelsize=26)\r\n        \r\n    positions = [0, np.pi\/4, np.pi\/2, 3*np.pi\/4, np.pi, 5*np.pi\/4, 3*np.pi\/2, 7*np.pi\/4]\r\n    for i,j in enumerate(ax.get_xticklabels()):\r\n        j.set_position((positions[i],-0.05))\r\n\r\nplt.show()<\/pre>\n
\"\"<\/p>\n
Temporal structure of synchrotron light<\/h3>\n
See Fig. 11.14.<\/p>\n
bbeta1 = 0.8\r\nbbeta2 = 0.4\r\nn_max = 250\r\nomega_0 = 10\r\nt = np.linspace(-0.2,3, 10000)\r\ns = omega_0*t\r\n\r\ncol = ['r', 'b']\r\n\r\nwith plt.style.context(('seaborn-v0_8-whitegrid')):\r\n    plt.rcParams.update(synchrotron_style)\r\n    fig = plt.figure(figsize=(9,6))\r\n    for i, bbeta in enumerate([0.4,0.8]):\r\n        aux = np.zeros_like(s)\r\n        for n in np.arange(-n_max,n_max,1):\r\n            aux = aux + n*np.exp(1j*n*s)*special.jvp(n, n*bbeta)\r\n    \r\n        if bbeta == 0.4:    \r\n            auxil = 20*np.abs(aux)**2    \r\n        else:\r\n            auxil = np.abs(aux)**2\r\n    \r\n        plt.plot(t,auxil, lw=2, color = col[i])\r\n    plt.show()<\/pre>\n
\"\"<\/p>\n
 <\/p>\n
 <\/p>\n","protected":false},"excerpt":{"rendered":"
Below is a set of python codes associated with Chapter 11 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":79,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"class_list":["post-808","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/808","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=808"}],"version-history":[{"count":3,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/808\/revisions"}],"predecessor-version":[{"id":828,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/808\/revisions\/828"}],"up":[{"embeddable":true,"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=\/wp\/v2\/pages\/79"}],"wp:attachment":[{"href":"https:\/\/synchrotron-light.net\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=808"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}