differt.em.L_i

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differt.em.L_i#

L_i(s_d, sin_2_beta_0, rho_1_i=None, rho_2_i=None, rho_e_i=None, s_i=None)[source]#

Compute the distance parameter associated with the incident shadow boundaries.

Note

This function can also be used to compute the distance parameters associated with the reflection shadow boundaries for the o- and n-faces, by passing the corresponding radii of curvature, see [11, eq. 6.28, p. 270].

Its general expression is given by [11, eq. 6.25, p. 270]:

\[L_i = \frac{(\rho_e^i + s)\rho_1^i\rho_2^i}{\rho_e^i(\rho_1^i + s)(\rho_2^i + s)}\sin^2\beta_0,\]

where \(s^d\) is the distance from the point of diffraction (\(Q_d\)) to the observer point (\(P\)), \(\rho_1^i\) is the principal radius of curvature of the incident wavefront at \(Q_d\) in the plane of incidence, \(\rho_2^i\) is the principal radius of curvature of the incident wavefront at \(Q_d\) in the plane transverse to the plane of incidence, \(\rho_e^i\) is radius of curvature of the incident wavefront in the edge-fixed plane of incidence., and \(\beta_0\) is the angle of diffraction.

By default, when \(\rho_1^i\), \(\rho_e^i\), and \(\rho_2^i\) are not provided, a plane wave incidence is assumed and the expression simplifies to [11, eq. 6.27, p. 270]:

\[L_i = s^d\sin^2\beta_0.\]

For spherical wavefront, you can pass \(s^i\) (s_i), the radius of curvature of the spherical wavefront, where \(s^i = \rho_1^i = \rho_2^2 = \rho_e^i\), and the expression will be simplified to [11, eq. 6.26, p. 270]:

\[L_i = \frac{s^ds^i}{s^d + s^i}\sin^2\beta_0.\]
Parameters:
Return type:

Float[Array, '*batch']

Returns:

The values of the distance parameter \(L_i\).

Raises:

ValueError – If ‘s_i’ was provided along at least one of the other radius parameters, or if one or the three ‘rho’ parameters was not provided.

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