Ming Xie formula¶

In [1]:

from zfel.mingxie import mingxie

from zfel.mingxie import mingxie

In [2]:

# Function signature
help(mingxie)

# Function signature help(mingxie)

Help on function mingxie in module zfel.mingxie:

mingxie(sigma_x=None, und_lambda=None, und_k=None, current=None, gamma=None, norm_emit=None, sigma_E=None)
    Calculates gain length using the Ming Xie formula.
    
    Design Optimization for an X-ray Free Elecgron Laser Driven by SLAC Linac
    Ming Xie, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA
    http://accelconf.web.cern.ch/AccelConf/p95/ARTICLES/TPG/TPG10.PDF
    
    Inputs keyword arguments:
        sigma_x      # RMS beam size
        und_lambda   # Undulator period (m)
        und_k        # Undulator K
        current      # Beam current (A)
        gamma        # Relativistic gamma
        norm_emit    # Normalized emittance (m-rad)
        sigma_E      # RMS energy spread (eV)
    
    
    Output as dict:
        gain_length        # Gain length (m)
        saturation_length  # Saturation length (m)
        saturation_power   # Saturation power (W)
        fel_wavelength     # FEL wavelength (m)
        pierce_parameter   # Pierce parameter (rho)

In [3]:

# Some test parameters
test_params = {    
    'sigma_x':18e-6,
    'und_lambda':0.03,
    'und_k':3.7,
    'current':5000.,
    'gamma':15000/0.511,
    'norm_emit':1e-6,
    'sigma_E':3e6}


mingxie(**test_params)

# Some test parameters test_params = { 'sigma_x':18e-6, 'und_lambda':0.03, 'und_k':3.7, 'current':5000., 'gamma':15000/0.511, 'norm_emit':1e-6, 'sigma_E':3e6} mingxie(**test_params)

Out[3]:

{'gain_length': 3.3615068621783646,
 'saturation_length': 61.57236107233048,
 'saturation_power': 27797096498.658634,
 'fel_wavelength': 1.3656628299999998e-10,
 'pierce_parameter': 0.0007257957842039326}

Vectorized version¶

The function will also work with inputs vectorized as numpy arrays. Here is an example scanning sigma_x

In [4]:

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

import numpy as np import matplotlib.pyplot as plt %matplotlib inline

In [5]:

n = 20
test_params2 = {}
# Duplicate all values
for k,v in test_params.items():
    test_params2[k] = np.array(n*[v])

# Vary sigma_x 
sx0 = test_params['sigma_x']
sx1 = 2*sx0
test_params2['sigma_x'] = np.linspace(sx0, sx1, n)    

# Calculate
result = mingxie(**test_params2)   
result

n = 20 test_params2 = {} # Duplicate all values for k,v in test_params.items(): test_params2[k] = np.array(n*[v]) # Vary sigma_x sx0 = test_params['sigma_x'] sx1 = 2*sx0 test_params2['sigma_x'] = np.linspace(sx0, sx1, n) # Calculate result = mingxie(**test_params2) result

Out[5]:

{'gain_length': array([3.36150686, 3.32585658, 3.30852171, 3.30590641, 3.31526357,
        3.33446805, 3.36185717, 3.39611655, 3.43619729, 3.48125509,
        3.53060491, 3.58368683, 3.64004011, 3.69928324, 3.76109848,
        3.82521978, 3.89142325, 3.95951962, 4.02934818, 4.10077195]),
 'saturation_length': array([61.57236107, 61.33146812, 61.36922496, 61.6335245 , 62.08397209,
        62.68891616, 63.42331198, 64.26716376, 65.20437429, 66.22188546,
        67.30902948, 68.45703405, 69.65864165, 70.90781394, 72.19950047,
        73.52945639, 74.8940976 , 76.29038502, 77.71573144, 79.16792625]),
 'saturation_power': array([2.77970965e+10, 2.93840283e+10, 3.06744364e+10, 3.16907401e+10,
        3.24599221e+10, 3.30105488e+10, 3.33707930e+10, 3.35672244e+10,
        3.36241540e+10, 3.35633548e+10, 3.34040247e+10, 3.31628954e+10,
        3.28544219e+10, 3.24910082e+10, 3.20832425e+10, 3.16401255e+10,
        3.11692818e+10, 3.06771522e+10, 3.01691629e+10, 2.96498750e+10]),
 'fel_wavelength': array([1.36566283e-10, 1.36566283e-10, 1.36566283e-10, 1.36566283e-10,
        1.36566283e-10, 1.36566283e-10, 1.36566283e-10, 1.36566283e-10,
        1.36566283e-10, 1.36566283e-10, 1.36566283e-10, 1.36566283e-10,
        1.36566283e-10, 1.36566283e-10, 1.36566283e-10, 1.36566283e-10,
        1.36566283e-10, 1.36566283e-10, 1.36566283e-10, 1.36566283e-10]),
 'pierce_parameter': array([0.0007258 , 0.0007014 , 0.00067895, 0.00065822, 0.000639  ,
        0.00062112, 0.00060445, 0.00058885, 0.00057421, 0.00056046,
        0.0005475 , 0.00053527, 0.00052369, 0.00051272, 0.00050231,
        0.00049241, 0.00048299, 0.000474  , 0.00046542, 0.00045722])}

In [6]:

# Plot
fig, ax = plt.subplots()
ax.set_xlabel('sigma_x (um)')
ax.set_ylabel('Gain length (m)')
ax.plot(1e6*test_params2['sigma_x'], result['gain_length'])

# Plot fig, ax = plt.subplots() ax.set_xlabel('sigma_x (um)') ax.set_ylabel('Gain length (m)') ax.plot(1e6*test_params2['sigma_x'], result['gain_length'])

Out[6]:

[<matplotlib.lines.Line2D at 0x7ff7519e2ee0>]

Contour plot¶

Reproduction of a plot similar to Figure 2 in the originam Ming Xie paper

In [7]:

i_x =20
i_y = 20
felwave = 1.5e-10
beta = 18
undL = 0.03
K = 3.7
sigmae = 3e6
I = np.linspace(2000, 8000, i_x)
norm_emit = np.linspace(0.5e-6, 2.5e-6, i_y);
gamma=np.sqrt(undL*(1+K**2/2)/2/felwave)
sigma_x = np.sqrt(beta*norm_emit/gamma)

test_params = {    
    'sigma_x':sigma_x[0],
    'und_lambda':undL,
    'und_k':K,
    'current':5000.,
    'gamma':gamma,
    'norm_emit':1e-6,
    'sigma_E':sigmae
}

X,Y = np.meshgrid(I, norm_emit)

contour = np.zeros((i_y, i_x))
for i in range(0, len(sigma_x)):
    test_params['current'] = X[i]
    test_params['norm_emit'] = Y[i]
    test_params['sigma_x'] = sigma_x[i]
    out = mingxie(**test_params)
    contour[i][:] = out['saturation_length']

fig, ax = plt.subplots()
c = ax.contour(X/1000, Y*1e6, contour, 25)
ax.clabel(c, inline=1, fontsize=10, fmt='%1.1f')
ax.set_title('Saturation Length (m)')
ax.set_ylabel('Normalized Emittance (mm-mrad)')
ax.set_xlabel('Current (kA)')

i_x =20 i_y = 20 felwave = 1.5e-10 beta = 18 undL = 0.03 K = 3.7 sigmae = 3e6 I = np.linspace(2000, 8000, i_x) norm_emit = np.linspace(0.5e-6, 2.5e-6, i_y); gamma=np.sqrt(undL*(1+K**2/2)/2/felwave) sigma_x = np.sqrt(beta*norm_emit/gamma) test_params = { 'sigma_x':sigma_x[0], 'und_lambda':undL, 'und_k':K, 'current':5000., 'gamma':gamma, 'norm_emit':1e-6, 'sigma_E':sigmae } X,Y = np.meshgrid(I, norm_emit) contour = np.zeros((i_y, i_x)) for i in range(0, len(sigma_x)): test_params['current'] = X[i] test_params['norm_emit'] = Y[i] test_params['sigma_x'] = sigma_x[i] out = mingxie(**test_params) contour[i][:] = out['saturation_length'] fig, ax = plt.subplots() c = ax.contour(X/1000, Y*1e6, contour, 25) ax.clabel(c, inline=1, fontsize=10, fmt='%1.1f') ax.set_title('Saturation Length (m)') ax.set_ylabel('Normalized Emittance (mm-mrad)') ax.set_xlabel('Current (kA)')

Out[7]:

Text(0.5, 0, 'Current (kA)')