Useful formulas

Statistics

twissed.weighted_avg(a: float | ndarray, weights: float | ndarray) → float | ndarray

Calculate the weighted average of array a.

\[\langle {\bf a} \rangle = \frac{\sum_i w_i a_i}{\sum_i w_i} \, ,\]

Parameters:

• arr (Union[float, np.ndarray]) – 1D numpy array

• weights (Union[float, np.ndarray]) – 1D numpy array

Returns:

weighted average.

Return type:

Union[float, np.ndarray]

twissed.weighted_std(arr: float | ndarray, weights: float | ndarray, verbose=True) → float | ndarray

Calculate the weighted standard deviation.

Parameters:

• arr (Union[float, np.ndarray]) – 1D numpy array

• weights (Union[float, np.ndarray]) – 1D numpy array

Returns:

weighted standard deviation.

Return type:

Union[float, np.ndarray]

twissed.weighted_med(arr: float | ndarray, weights: float | ndarray, verbose=True) → float | ndarray

Compute the weighted median

Parameters:

• a (Union[float, np.ndarray]) – 1D numpy array

• weights (Union[float, np.ndarray]) – 1D numpy array

Returns:

weighted median

Return type:

Union[float, np.ndarray]

twissed.weighted_mad(a, w)

Compute the weighted median absolute

Parameters:

• a (float) – 1D numpy array

• weights (float) – 1D numpy array

Returns:

median

Return type:

float

Physics

twissed.laser_strength(I0: float, lambda0: float, normalised: bool = False) → float

Laser strength parameter \(a_0\).

\[a_0 = \sqrt{\frac{e^2}{2 \pi^2 \epsilon_0 m_e^2 c^5} \lambda_0 I_0}\]

or if normalised == True:

\[a_0 = 0.855 \lambda_0 [\mu \mathrm{m}] \sqrt{I_0 [10^{18} \mathrm{W/cm}^2]}\]

Parameters:

• I0 (float) – Maximum intensity

• lambda0 (float) – Wavelength

• normalised (bool) – Normalised equation. Default to False.

Returns:

Laser strength parameter

Return type:

float

twissed.critical_density(lambda0: float) → float

Laser critical density \(n_{\mathrm{c}}\).

\[n_{\mathrm{c}} = \frac{m_e \epsilon_0 \omega_0^2}{e^2}\]

Parameters:

lambda0 (float) – Wavelength

Returns:

Laser critical density

Return type:

float

twissed.Rayleigh_length(w0: float, lambda0: float) → float

Return the Rayleigh length \(z_{\rm R}\).

\[z_{\rm R} = \frac{\pi w_0^2}{\lambda_0}\]

Parameters:

• w0 (float) – Maximum waist

• lambda0 (float) – Wavelength

Returns:

Rayleigh length

Return type:

float

twissed.accelerating_electric_field(ne: float) → float

Return the maximum accelerating electric field (a.k.a the wavebreaking field or space charge field) \(E_{\rm max}\).

\[E_{\rm max} = \frac{m_e c \omega_{\rm pe}}{e}\]

Parameters:

ne (float) – Plasma density in m-3

Returns:

Maximum accelerating electric field in V/m

Return type:

float

twissed.waist0_theory(z: float | ndarray, w0: float, lambda0: float, zfoc: float = 0) → float | ndarray

Return the theoretical waist over \(z\).

\[w(z) = w_0 \sqrt{1 + \left( \frac{z - z_{\rm foc}}{z_{\rm R}} \right)^2}\]

Parameters:

• z (Union[float, np.ndarray]) – Positions in z

• w0 (float) – Maximum waist

• zfoc (float) – Z focal position. Default to 0.

• lambda0 (float) – Laser wavelength

Returns:

Theoretical waist

Return type:

Union[float, np.ndarray]

twissed.plasma_frequency(ne: float) → float

Return the theoretical waist over \(z\).

\[\omega_{\rm pe} = \sqrt{\frac{n_{\rm e} e^2}{m_{\rm e} \epsilon_0}}\]

Parameters:

ne (float) – Plasma density in m-3

Returns:

Plasma frequency in rad/s.

Return type:

float

Helpful

twissed.convert_laser_duration_FWHM(t: float, convert: str = 'fbpic_to_FWHM') → float

Convert laser FWHM duration.

Parameters:

• t (float) – time to convert.

• convert (str, optional) – Type of conversion. Chose between “fbpic_to_FWHM”, “FWHM_to_fbpic”, “smilei_to_FWHM” or “FWHM_to_smilei”. Defaults to “fbpic_to_FWHM”.

Returns:

Converted time

Return type:

float