Module wsp_tools.beam
Module to generate spatial modes and calculate beam parameters.
Expand source code
# wsp-tools is TEM data analysis and simulation tools developed by WSP as a grad student in the McMorran Lab.
# Copyright (C) 2021 William S. Parker
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
"""Module to generate spatial modes and calculate beam parameters.
"""
import numpy as np
from . import constants as _
from scipy.special import eval_genlaguerre, factorial
__all__ = ['energy','p','dB','k','omega','v_g','v_p','bessel','besselPacket','zR','w','roc','LG']
def energy(T_eV):
"""Calculates the total electron energy from its kinetic energy.
**Parameters**
* **T_eV** : _number_ <br />
the kinetic energy in eV.
**Returns**
* **energy(T_eV)** : _number_ <br />
the total energy in eV.
"""
return(T_eV * _.e + _.m_e * _.c**2)
def p(T_eV):
"""Calculates the momentum from the electron's kinetic energy.
**Parameters**
* **T_eV** : _number_ <br />
the kinetic energy in eV.
**Returns**
* **p(T_eV)** : _number_ <br />
the momentum in N s.
"""
return(1/_.c * np.sqrt(energy(T_eV)**2 - (_.m_e * _.c**2)**2))
def dB(T_eV):
"""Calculates the de Broglie wavelength from the electron's kinetic energy.
**Parameters**
* **T_eV** : _number_ <br />
the kinetic energy in eV.
**Returns**
* **dB(T_eV)** : _number_ <br />
the de Broglie wavelength in m.
"""
return(_.h / p(T_eV))
def k(T_eV):
"""Calculates the wavenumber from the electron's kinetic energy.
**Parameters**
* **T_eV** : _number_ <br />
the kinetic energy in eV.
**Returns**
* **k(T_eV)** : _number_ <br />
the wavenumber in m<sup>-1</sup>.
"""
return(p(T_eV) / _.hbar)
def omega(T_eV):
"""Calculates the angular frequency from the electron's kinetic energy.
**Parameters**
* **T_eV** : _number_ <br />
the kinetic energy in eV.
**Returns**
* **omega(T_eV)** : _number_ <br />
the angular frequency in rad s<sup>-1</sup>.
"""
return(energy(T_eV) / _.hbar)
def v_g(T_eV):
"""Calculates the group velocity from the electron's kinetic energy.
**Parameters**
* **T_eV** : _number_ <br />
the kinetic energy in eV.
**Returns**
* **v_g(T_eV)** : _number <br />
the group velocity in m s<sup>-1</sup>.
"""
return(_.c * k(T_eV) / np.sqrt(_.m_e**2 * _.c**2 / _.hbar**2 + k(T_eV)**2))
def v_p(T_eV):
"""Calculates the phase velocity from the electron's kinetic energy.
**Parameters**
* **T_eV** : _number_ <br />
the kinetic energy in eV.
**Returns**
* **v_p(T_eV)** : _number_ <br />
the phase velocity in m s<sup>-1</sup>.
"""
return(omega(T_eV)/k(T_eV))
def bessel(x, y, z = 0, l = 0, kz0 = k(3e5),
kperp = 0.0005*k(3e5), dkperp = 0, N = 30):
"""Creates a Bessel beam by adding plane waves.
The spectrum is a circle in k-space <br />`(kz0, dkperp + kperp * cos(theta), kperp * sin(theta))`.
**Parameters**
* **x** : _number, ndarray_ <br />
the x-coordinates over which to calculate the beam.
* **y** : _number, ndarray_ <br />
the y-coordinates over which to calculate the beam.
* **z** : _number, ndarray, optional_ <br />
the z-coordinates over which to calculate the beam. <br />
Default is `z = 0`.
* **l** : _number, optional_ <br />
the winding number, or chirality, of the beam. <br />
Default is `l = 0`.
* **kz0** : _number, optional_ <br />
the z-coordinate of the spectrum. That is, the z component of the beam's wavevector. <br />
Default is `kz0 = k(3e5)`.
* **kperp** : _number, optional_ <br />
the perpendicular component of the spectrum. That is, the perpendicular component of the beam's wavevector. This is the radius of the beam's spectrum. <br />
Default is `kperp = 5e-4 * k(3e5)`.
* **dkperp** : _number, optional_ <br />
the perpendicular offset of the spectrum. This is applied in the `k_x` direction. <br />
Default is `dkperp = 0`.
* **N** : _int, optional_ <br />
The number of plane waves to use. <br />
Default is `N = 30`.
**Returns**
* **mode** : _ndarray_ <br />
The complex amplitudes of the Bessel beam at the specified x, y, and z positions.
Shape is broadcast between x, y, and z inputs.
"""
mode = x*y*z*0j
w_0 = np.sqrt((kz0**2 + kperp**2)*_.c**2 + _.m_e**2*_.c**4/_.hbar**2)
for n in range(N):
phi_n = 2*np.pi*n/N
kxn, kyn, kzn = kperp*np.cos(phi_n) + dkperp, kperp*np.sin(phi_n), kz0
w_n = np.sqrt(w_0**2 + dkperp**2 * _.c**2 + 2*kxn*dkperp*_.c**2)
mode += np.exp(1j*(kxn*x + kyn*y + kzn*z + l*phi_n))
mode *= 1/np.sqrt(N)
return(mode)
def besselPacket(t = 0, l = 0,
kres = 2**7, kmin = -3*k(3e5), kmax = 3*k(3e5),
kz0 = k(3e5), kperp = .5*k(3e5), dkperp = 0,
sig = 0.05*k(3e5)):
"""Creates a bessel beam by Fourier transforming a Gaussian spectrum.
The spectrum is a gaussian centered on a circle in
k-space <br /> `(k_z, dkperp + kperp * cos(theta), kperp * sin(theta))`.
**Parameters**
* **t** : _number, optional_ <br />
time in seconds. <br />
Default is `t = 0`.
* **l** : _number, optional_ <br />
the winding number, or chirality, of the beam. <br />
Default is `l = 0`.
* **kres** : _int, optional_ <br />
the resolution of the k-space; also, the resolution of the output beam. Note that the function will be faster if the resolution is a power of 2. <br />
Default is `kres = 128`.
* **kmin** : _number, optional_ <br />
the minimum value in k-space. This is applied to the x, y, and z components of the wavenumber. <br />
Default is `kmin = -3 * k(3e5)`.
* **kmax** : _number, optional_ <br />
the maximum value in k-space. This is applied to the x, y, and z components of the wavenumber. <br />
Default is `kmax = 3 * k(3e5)`.
* **kz0** : _number, optional_ <br />
the z-coordinate of the spectrum. That is, the z component of the beam's wavevector. <br />
Default is `kz0 = k(3e5)`.
* **kperp** : _number, optional_ <br />
the perpendicular component of the spectrum. That is, the perpendicular component of the beam's wavevector. This is the radius of the beam's spectrum. <br />
Default is `kperp = 5e-4 * k(3e5)`.
* **dkperp** : _number, optional_ <br />
the perpendicular offset of the spectrum. This is applied in the `k_x` direction. <br />
Default is `dkperp = 0`.
* **sig** : _number, optional_ <br />
the standard deviation of the Gaussian envelope around the circle in k-space. <br />
Default is `sig = 0.05 * k(3e5)`.
**Returns**
* **mode** : _ndarray_ <br />
three-dimensional array containing the complex amplitudes of the Bessel packet. Shape is `kres`x`kres`x`kres`.
"""
# K = np.linspace(kmin, kmax, kres)
kx, ky, kz = np.ogrid[kmin:kmax:1j*kres,kmin:kmax:1j*kres,kmin:kmax:1j*kres]
w = _.c * np.sqrt((kx**2 + ky**2 + kz**2) + _.c**2 * _.m_e**2 /_.hbar**2)
# cylindrical
kr = np.sqrt(kx**2 + ky**2)
mode = kz*0j
phi = np.arctan2(ky,kx)
spec = np.exp( -( (kr-kperp)**2 + (kz-kz0)**2 ) / (2*sig**2)) \
* np.exp(1j*phi*l-1j*w*t)
### Include first ifftshift because spec is [-k0, k0], not [0,k0]
mode = np.fft.fftshift(np.fft.ifftn(np.fft.ifftshift(spec)))
return(mode)
def zR(k, w0):
"""Rayleigh range as a function of wavenumber and beam waist.
**Parameters**
* **k** : _number_ <br />
Wavenumber.
* **w0** : _number_ <br />
Beam waist.
**Returns**
* **zR(k, w0)** : _number_ <br />
the Rayleigh range.
"""
lam = 2*_.pi/k
return(_.pi * w0**2 / lam)
def w(z,w0,k):
"""Spot size as a function of z, beam waist, and wavenumber.
**Parameters**
* **z** : _number_ <br />
z-position.
* **w0** : _number_ <br />
beam waist.
* **k** : _number_ <br />
wavenumber.
**Returns**
* **w(z, w0, k)** : _number_ <br />
the spot size at z.
"""
return(w0 * np.sqrt(1 + (z/zR(k,w0))**2))
np.seterr(divide='ignore')
def roc(z, w0, k):
"""Radius of curvature as a function of z, beam waist, and wavenumber.
**Parameters**
* **z** : _number_ <br />
z-position.
* **w0** : _number_ <br />
beam waist.
* **k** : _number_ <br />
wavenumber.
**Returns**
* **roc(z, w0, k)** : _number_ <br />
Radius of curvature.
"""
return(z + np.divide(zR(k,w0)**2,z))
def LG(x, y, z = 0, l = 0, p = 0, w_0 = 2e-6, lam = 1.97e-12):
"""Generates a Laguerre-Gauss spatial mode.
Note: broadcasting is not supported - if x and y are both 1d arrays, the
result will be a 1darray and will NOT make sense. Likewise if x, y are 2d
arrays, but z is a 1d array, the result will be an error.
**Parameters**
* **x** : _number, ndarray_ <br />
The x-coordinates over which to calculate the beam.
* **y** : _number, ndarray_ <br />
The y-coordinates over which to calculate the beam.
* **z** : _number, ndarray, optional_ <br />
The z-coordinates over which to calculate the beam. <br />
Default is `z = 0`.
* **l** : _number, optional_ <br />
The winding number, or chirality, of the beam. <br />
Default is `l = 0`.
* **p** : _number, optional_ <br />
The radial index of the beam. <br />
Default is `p = 0`.
* **w_0** : _number, optional_ <br />
The beam waist. <br />
Default is `w_0 = 2e-6`.
* **lam** : _number, optional_ <br />
The beam's wavelength. <br />
Default is `lam = 1.97e-12` (the relativistic wavelength of a 300keV electron).
**Returns**
* **mode** : _ndarray_ <br />
The complex amplitudes of a Laguerre Gaussian beam at the specified x, y, and z positions.
"""
r, theta = np.sqrt(x**2 + y**2), np.arctan2(y,x)
Clp = np.sqrt(2*factorial(p)/np.pi/factorial(p+np.abs(l)))
z_r = np.pi*w_0**2/lam
w_z = w_0*np.sqrt(1 + z**2/z_r**2)
gouy = (2*p + np.abs(l) + 1) * np.nan_to_num(np.arctan2(z,z_r))
r = np.sqrt(x**2+y**2); theta = np.arctan2(y,x)
mode = Clp \
* w_0/w_z \
* (r * np.sqrt(2)/w_z)**np.abs(l) \
* np.exp(-r**2/w_z**2) \
* eval_genlaguerre(p,np.abs(l),(2*r**2)/w_z**2) \
* np.exp(-1j * 2 * np.pi / lam * r**2 * z/ 2 /(z**2 + z_r**2)) \
* np.exp(-1j * l * theta) \
* np.exp(1j * (1 + np.abs(l) * 2*p))
mode /= np.max(np.abs(mode))
return(mode)Functions
def LG(x, y, z=0, l=0, p=0, w_0=2e-06, lam=1.97e-12)-
Generates a Laguerre-Gauss spatial mode.
Note: broadcasting is not supported - if x and y are both 1d arrays, the result will be a 1darray and will NOT make sense. Likewise if x, y are 2d arrays, but z is a 1d array, the result will be an error.
Parameters
-
x : number, ndarray
The x-coordinates over which to calculate the beam. -
y : number, ndarray
The y-coordinates over which to calculate the beam. -
z : number, ndarray, optional
The z-coordinates over which to calculate the beam.
Default isz = 0. -
l : number, optional
The winding number, or chirality, of the beam.
Default isl = 0. -
p : number, optional
The radial index of the beam.
Default isp = 0. -
w_0 : number, optional
The beam waist.
Default isw_0 = 2e-6. -
lam : number, optional
The beam's wavelength.
Default islam = 1.97e-12(the relativistic wavelength of a 300keV electron).
Returns
- mode : ndarray
The complex amplitudes of a Laguerre Gaussian beam at the specified x, y, and z positions.
Expand source code
def LG(x, y, z = 0, l = 0, p = 0, w_0 = 2e-6, lam = 1.97e-12): """Generates a Laguerre-Gauss spatial mode. Note: broadcasting is not supported - if x and y are both 1d arrays, the result will be a 1darray and will NOT make sense. Likewise if x, y are 2d arrays, but z is a 1d array, the result will be an error. **Parameters** * **x** : _number, ndarray_ <br /> The x-coordinates over which to calculate the beam. * **y** : _number, ndarray_ <br /> The y-coordinates over which to calculate the beam. * **z** : _number, ndarray, optional_ <br /> The z-coordinates over which to calculate the beam. <br /> Default is `z = 0`. * **l** : _number, optional_ <br /> The winding number, or chirality, of the beam. <br /> Default is `l = 0`. * **p** : _number, optional_ <br /> The radial index of the beam. <br /> Default is `p = 0`. * **w_0** : _number, optional_ <br /> The beam waist. <br /> Default is `w_0 = 2e-6`. * **lam** : _number, optional_ <br /> The beam's wavelength. <br /> Default is `lam = 1.97e-12` (the relativistic wavelength of a 300keV electron). **Returns** * **mode** : _ndarray_ <br /> The complex amplitudes of a Laguerre Gaussian beam at the specified x, y, and z positions. """ r, theta = np.sqrt(x**2 + y**2), np.arctan2(y,x) Clp = np.sqrt(2*factorial(p)/np.pi/factorial(p+np.abs(l))) z_r = np.pi*w_0**2/lam w_z = w_0*np.sqrt(1 + z**2/z_r**2) gouy = (2*p + np.abs(l) + 1) * np.nan_to_num(np.arctan2(z,z_r)) r = np.sqrt(x**2+y**2); theta = np.arctan2(y,x) mode = Clp \ * w_0/w_z \ * (r * np.sqrt(2)/w_z)**np.abs(l) \ * np.exp(-r**2/w_z**2) \ * eval_genlaguerre(p,np.abs(l),(2*r**2)/w_z**2) \ * np.exp(-1j * 2 * np.pi / lam * r**2 * z/ 2 /(z**2 + z_r**2)) \ * np.exp(-1j * l * theta) \ * np.exp(1j * (1 + np.abs(l) * 2*p)) mode /= np.max(np.abs(mode)) return(mode) -
def bessel(x, y, z=0, l=0, kz0=3191460985702.0464, kperp=1595730492.8510232, dkperp=0, N=30)-
Creates a Bessel beam by adding plane waves.
The spectrum is a circle in k-space
(kz0, dkperp + kperp * cos(theta), kperp * sin(theta)).Parameters
-
x : number, ndarray
the x-coordinates over which to calculate the beam. -
y : number, ndarray
the y-coordinates over which to calculate the beam. -
z : number, ndarray, optional
the z-coordinates over which to calculate the beam.
Default isz = 0. -
l : number, optional
the winding number, or chirality, of the beam.
Default isl = 0. -
kz0 : number, optional
the z-coordinate of the spectrum. That is, the z component of the beam's wavevector.
Default iskz0 = k(3e5). -
kperp : number, optional
the perpendicular component of the spectrum. That is, the perpendicular component of the beam's wavevector. This is the radius of the beam's spectrum.
Default iskperp = 5e-4 * k(3e5). -
dkperp : number, optional
the perpendicular offset of the spectrum. This is applied in thek_xdirection.
Default isdkperp = 0. -
N : int, optional
The number of plane waves to use.
Default isN = 30.
Returns
- mode : ndarray
The complex amplitudes of the Bessel beam at the specified x, y, and z positions. Shape is broadcast between x, y, and z inputs.
Expand source code
def bessel(x, y, z = 0, l = 0, kz0 = k(3e5), kperp = 0.0005*k(3e5), dkperp = 0, N = 30): """Creates a Bessel beam by adding plane waves. The spectrum is a circle in k-space <br />`(kz0, dkperp + kperp * cos(theta), kperp * sin(theta))`. **Parameters** * **x** : _number, ndarray_ <br /> the x-coordinates over which to calculate the beam. * **y** : _number, ndarray_ <br /> the y-coordinates over which to calculate the beam. * **z** : _number, ndarray, optional_ <br /> the z-coordinates over which to calculate the beam. <br /> Default is `z = 0`. * **l** : _number, optional_ <br /> the winding number, or chirality, of the beam. <br /> Default is `l = 0`. * **kz0** : _number, optional_ <br /> the z-coordinate of the spectrum. That is, the z component of the beam's wavevector. <br /> Default is `kz0 = k(3e5)`. * **kperp** : _number, optional_ <br /> the perpendicular component of the spectrum. That is, the perpendicular component of the beam's wavevector. This is the radius of the beam's spectrum. <br /> Default is `kperp = 5e-4 * k(3e5)`. * **dkperp** : _number, optional_ <br /> the perpendicular offset of the spectrum. This is applied in the `k_x` direction. <br /> Default is `dkperp = 0`. * **N** : _int, optional_ <br /> The number of plane waves to use. <br /> Default is `N = 30`. **Returns** * **mode** : _ndarray_ <br /> The complex amplitudes of the Bessel beam at the specified x, y, and z positions. Shape is broadcast between x, y, and z inputs. """ mode = x*y*z*0j w_0 = np.sqrt((kz0**2 + kperp**2)*_.c**2 + _.m_e**2*_.c**4/_.hbar**2) for n in range(N): phi_n = 2*np.pi*n/N kxn, kyn, kzn = kperp*np.cos(phi_n) + dkperp, kperp*np.sin(phi_n), kz0 w_n = np.sqrt(w_0**2 + dkperp**2 * _.c**2 + 2*kxn*dkperp*_.c**2) mode += np.exp(1j*(kxn*x + kyn*y + kzn*z + l*phi_n)) mode *= 1/np.sqrt(N) return(mode) -
def besselPacket(t=0, l=0, kres=128, kmin=-9574382957106.139, kmax=9574382957106.139, kz0=3191460985702.0464, kperp=1595730492851.0232, dkperp=0, sig=159573049285.10233)-
Creates a bessel beam by Fourier transforming a Gaussian spectrum.
The spectrum is a gaussian centered on a circle in k-space
(k_z, dkperp + kperp * cos(theta), kperp * sin(theta)).Parameters
-
t : number, optional
time in seconds.
Default ist = 0. -
l : number, optional
the winding number, or chirality, of the beam.
Default isl = 0. -
kres : int, optional
the resolution of the k-space; also, the resolution of the output beam. Note that the function will be faster if the resolution is a power of 2.
Default iskres = 128. -
kmin : number, optional
the minimum value in k-space. This is applied to the x, y, and z components of the wavenumber.
Default iskmin = -3 * k(3e5). -
kmax : number, optional
the maximum value in k-space. This is applied to the x, y, and z components of the wavenumber.
Default iskmax = 3 * k(3e5). -
kz0 : number, optional
the z-coordinate of the spectrum. That is, the z component of the beam's wavevector.
Default iskz0 = k(3e5). -
kperp : number, optional
the perpendicular component of the spectrum. That is, the perpendicular component of the beam's wavevector. This is the radius of the beam's spectrum.
Default iskperp = 5e-4 * k(3e5). -
dkperp : number, optional
the perpendicular offset of the spectrum. This is applied in thek_xdirection.
Default isdkperp = 0. -
sig : number, optional
the standard deviation of the Gaussian envelope around the circle in k-space.
Default issig = 0.05 * k(3e5).
Returns
- mode : ndarray
three-dimensional array containing the complex amplitudes of the Bessel packet. Shape iskresxkresxkres.
Expand source code
def besselPacket(t = 0, l = 0, kres = 2**7, kmin = -3*k(3e5), kmax = 3*k(3e5), kz0 = k(3e5), kperp = .5*k(3e5), dkperp = 0, sig = 0.05*k(3e5)): """Creates a bessel beam by Fourier transforming a Gaussian spectrum. The spectrum is a gaussian centered on a circle in k-space <br /> `(k_z, dkperp + kperp * cos(theta), kperp * sin(theta))`. **Parameters** * **t** : _number, optional_ <br /> time in seconds. <br /> Default is `t = 0`. * **l** : _number, optional_ <br /> the winding number, or chirality, of the beam. <br /> Default is `l = 0`. * **kres** : _int, optional_ <br /> the resolution of the k-space; also, the resolution of the output beam. Note that the function will be faster if the resolution is a power of 2. <br /> Default is `kres = 128`. * **kmin** : _number, optional_ <br /> the minimum value in k-space. This is applied to the x, y, and z components of the wavenumber. <br /> Default is `kmin = -3 * k(3e5)`. * **kmax** : _number, optional_ <br /> the maximum value in k-space. This is applied to the x, y, and z components of the wavenumber. <br /> Default is `kmax = 3 * k(3e5)`. * **kz0** : _number, optional_ <br /> the z-coordinate of the spectrum. That is, the z component of the beam's wavevector. <br /> Default is `kz0 = k(3e5)`. * **kperp** : _number, optional_ <br /> the perpendicular component of the spectrum. That is, the perpendicular component of the beam's wavevector. This is the radius of the beam's spectrum. <br /> Default is `kperp = 5e-4 * k(3e5)`. * **dkperp** : _number, optional_ <br /> the perpendicular offset of the spectrum. This is applied in the `k_x` direction. <br /> Default is `dkperp = 0`. * **sig** : _number, optional_ <br /> the standard deviation of the Gaussian envelope around the circle in k-space. <br /> Default is `sig = 0.05 * k(3e5)`. **Returns** * **mode** : _ndarray_ <br /> three-dimensional array containing the complex amplitudes of the Bessel packet. Shape is `kres`x`kres`x`kres`. """ # K = np.linspace(kmin, kmax, kres) kx, ky, kz = np.ogrid[kmin:kmax:1j*kres,kmin:kmax:1j*kres,kmin:kmax:1j*kres] w = _.c * np.sqrt((kx**2 + ky**2 + kz**2) + _.c**2 * _.m_e**2 /_.hbar**2) # cylindrical kr = np.sqrt(kx**2 + ky**2) mode = kz*0j phi = np.arctan2(ky,kx) spec = np.exp( -( (kr-kperp)**2 + (kz-kz0)**2 ) / (2*sig**2)) \ * np.exp(1j*phi*l-1j*w*t) ### Include first ifftshift because spec is [-k0, k0], not [0,k0] mode = np.fft.fftshift(np.fft.ifftn(np.fft.ifftshift(spec))) return(mode) -
def dB(T_eV)-
Calculates the de Broglie wavelength from the electron's kinetic energy.
Parameters
- T_eV : number
the kinetic energy in eV.
Returns
- dB(T_eV) : number
the de Broglie wavelength in m.
Expand source code
def dB(T_eV): """Calculates the de Broglie wavelength from the electron's kinetic energy. **Parameters** * **T_eV** : _number_ <br /> the kinetic energy in eV. **Returns** * **dB(T_eV)** : _number_ <br /> the de Broglie wavelength in m. """ return(_.h / p(T_eV)) - T_eV : number
def energy(T_eV)-
Calculates the total electron energy from its kinetic energy.
Parameters
- T_eV : number
the kinetic energy in eV.
Returns
- energy(T_eV) : number
the total energy in eV.
Expand source code
def energy(T_eV): """Calculates the total electron energy from its kinetic energy. **Parameters** * **T_eV** : _number_ <br /> the kinetic energy in eV. **Returns** * **energy(T_eV)** : _number_ <br /> the total energy in eV. """ return(T_eV * _.e + _.m_e * _.c**2) - T_eV : number
def k(T_eV)-
Calculates the wavenumber from the electron's kinetic energy.
Parameters
- T_eV : number
the kinetic energy in eV.
Returns
- k(T_eV) : number
the wavenumber in m-1.
Expand source code
def k(T_eV): """Calculates the wavenumber from the electron's kinetic energy. **Parameters** * **T_eV** : _number_ <br /> the kinetic energy in eV. **Returns** * **k(T_eV)** : _number_ <br /> the wavenumber in m<sup>-1</sup>. """ return(p(T_eV) / _.hbar) - T_eV : number
def omega(T_eV)-
Calculates the angular frequency from the electron's kinetic energy.
Parameters
- T_eV : number
the kinetic energy in eV.
Returns
- omega(T_eV) : number
the angular frequency in rad s-1.
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def omega(T_eV): """Calculates the angular frequency from the electron's kinetic energy. **Parameters** * **T_eV** : _number_ <br /> the kinetic energy in eV. **Returns** * **omega(T_eV)** : _number_ <br /> the angular frequency in rad s<sup>-1</sup>. """ return(energy(T_eV) / _.hbar) - T_eV : number
def p(T_eV)-
Calculates the momentum from the electron's kinetic energy.
Parameters
- T_eV : number
the kinetic energy in eV.
Returns
- p(T_eV) : number
the momentum in N s.
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def p(T_eV): """Calculates the momentum from the electron's kinetic energy. **Parameters** * **T_eV** : _number_ <br /> the kinetic energy in eV. **Returns** * **p(T_eV)** : _number_ <br /> the momentum in N s. """ return(1/_.c * np.sqrt(energy(T_eV)**2 - (_.m_e * _.c**2)**2)) - T_eV : number
def roc(z, w0, k)-
Radius of curvature as a function of z, beam waist, and wavenumber.
Parameters
-
z : number
z-position. -
w0 : number
beam waist. -
k : number
wavenumber.
Returns
- roc(z, w0, k) : number
Radius of curvature.
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def roc(z, w0, k): """Radius of curvature as a function of z, beam waist, and wavenumber. **Parameters** * **z** : _number_ <br /> z-position. * **w0** : _number_ <br /> beam waist. * **k** : _number_ <br /> wavenumber. **Returns** * **roc(z, w0, k)** : _number_ <br /> Radius of curvature. """ return(z + np.divide(zR(k,w0)**2,z)) -
def v_g(T_eV)-
Calculates the group velocity from the electron's kinetic energy.
Parameters
- T_eV : number
the kinetic energy in eV.
Returns
- v_g(T_eV) : _number
the group velocity in m s-1.
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def v_g(T_eV): """Calculates the group velocity from the electron's kinetic energy. **Parameters** * **T_eV** : _number_ <br /> the kinetic energy in eV. **Returns** * **v_g(T_eV)** : _number <br /> the group velocity in m s<sup>-1</sup>. """ return(_.c * k(T_eV) / np.sqrt(_.m_e**2 * _.c**2 / _.hbar**2 + k(T_eV)**2)) - T_eV : number
def v_p(T_eV)-
Calculates the phase velocity from the electron's kinetic energy.
Parameters
- T_eV : number
the kinetic energy in eV.
Returns
- v_p(T_eV) : number
the phase velocity in m s-1.
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def v_p(T_eV): """Calculates the phase velocity from the electron's kinetic energy. **Parameters** * **T_eV** : _number_ <br /> the kinetic energy in eV. **Returns** * **v_p(T_eV)** : _number_ <br /> the phase velocity in m s<sup>-1</sup>. """ return(omega(T_eV)/k(T_eV)) - T_eV : number
def w(z, w0, k)-
Spot size as a function of z, beam waist, and wavenumber.
Parameters
-
z : number
z-position. -
w0 : number
beam waist. -
k : number
wavenumber.
Returns
- w(z, w0, k) : number
the spot size at z.
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def w(z,w0,k): """Spot size as a function of z, beam waist, and wavenumber. **Parameters** * **z** : _number_ <br /> z-position. * **w0** : _number_ <br /> beam waist. * **k** : _number_ <br /> wavenumber. **Returns** * **w(z, w0, k)** : _number_ <br /> the spot size at z. """ return(w0 * np.sqrt(1 + (z/zR(k,w0))**2)) -
def zR(k, w0)-
Rayleigh range as a function of wavenumber and beam waist.
Parameters
-
k : number
Wavenumber. -
w0 : number
Beam waist.
Returns
- zR(k, w0) : number
the Rayleigh range.
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def zR(k, w0): """Rayleigh range as a function of wavenumber and beam waist. **Parameters** * **k** : _number_ <br /> Wavenumber. * **w0** : _number_ <br /> Beam waist. **Returns** * **zR(k, w0)** : _number_ <br /> the Rayleigh range. """ lam = 2*_.pi/k return(_.pi * w0**2 / lam) -