focusedBeam

int — maximum total angular momentum quantum number.

float — wavelength of the incident beam (in microns).

domain_class object — spatial domain for field evaluation.

int — helicity of the beam before focusing (-1 or 1, default is 1).

int — azimuthal number of the beam (default is 0).

int — radial index of the beam (default is 0).

float — numerical aperture of the focusing lens (default is 0.9). Allowed values are \(0.25, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 \& 0.9\).

float — focal length of the lens (in microns, default is 1000).

float — refractive index of the lens medium (default is 1).

Computes the Wigner small-d function \(d_{m p}^j(\Theta)\) for given quantum numbers \(j\), \(m\), and \(p\) at angle \(\Theta\). The function is defined as: \[\begin{aligned} d_{m p}^j(\Theta) = (-1)^{\frac{p - m - |m - p|}{2}} &\exp\left( \frac{1}{2} \left[ \ln \Gamma(j - M + 1) + \ln \Gamma(j + M + 1) \right.\right. \\ &\left.\left.- \ln \Gamma(j + N + 1) - \ln \Gamma(j - N + 1) \right] \right) \\ &\cdot \cos\left(\frac{\Theta}{2}\right)^{|m + p|} \sin\left(\frac{\Theta}{2}\right)^{|m - p|} P_{j - M}^{|m - p|, |m + p|}(\cos \Theta), \end{aligned}\] where \(M = \max(|m|, |p|)\), \(N = \min(|m|, |p|)\), and \(P_n^{\alpha, \beta}\) is the Jacobi polynomial evaluated using scipy.special.eval_jacobi.

Computes the beam coefficients \(C_{j m_z p}\) for the Laguerre-Gaussian beam with quantum numbers \(l\), \(p\), and \(q\). The coefficient is calculated via numerical integration over the lens aperture: \[\label{eq:Cjmz} C_{j m_z p} = \int_0^{\theta_{\text{max}}} \sin\theta \, f e^{-i k f} \sqrt{2\pi} \sqrt{n_{\text{lens}} \cos\theta} \, \text{LG}(q, l,\theta) \, d_{m_z p}^j(\theta) \, d\theta,\] where \(m_z = l + p\), \(\theta_{\text{max}} = \arcsin(\text{NA} / n_{\text{lens}})\), \(f\) is the focal length, \(k = 2\pi / \lambda\), and \(\text{LG}(q,l,\theta)\) is the Laguerre-Gaussian beam amplitude defined with another method. Returns the coefficients \(C\), the lens integral (which should be as close to unity as possible), and the normalization sum \(\sum_j (2j + 1) |C_j|^2\), ideally close to 2(Zambrana-Puyalto 2014).

Visualizes the squared magnitude of beam coefficients \(|C_{j m_z p}|^2\) as a bar plot for specified \(l\), \(p\), and \(q\). The plot displays contributions for \(j\) from \(\max(|m_z|, 1)\) to maxJ, with optional multiple parameter combinations.

Computes the Laguerre-Gaussian beam amplitude at radial distance \(\rho\) and axial position \(z\). The amplitude is given by: \[\begin{aligned} \text{LG} = \exp\bigg( \ln N(l, q) &- \frac{\rho^2}{w^2} + l \ln \rho + (l + 1) \left( \frac{1}{2} \ln 2 - \ln w \right) \bigg) L_q^l\left( \frac{2 \rho^2}{w^2} \right), \end{aligned}\] where \(N(l, q) = \sqrt{\frac{q!}{\pi (q + l)!}}\) is the normalization factor, \(w\) is the beam waist, \(k = 2\pi / \lambda\), and \(L_q^l\) is the generalized Laguerre polynomial.

Returns the beam waist \(w\) (in microns) for a given numerical aperture NA and orbital quantum number \(l\), based on precomputed values for NA = 0.25, 0.3, 0.5, or 0.9. Beam waist values are computed such that they fill the lens completely and the sum is truncated appropriately(Zambrana-Puyalto 2014).

Computes the total electric field for the focused beam by summing electric and magnetic contributions: \[E = \sum_{j=j_0}^{j_{\text{max}}} i^j \sqrt{2j + 1} C_j \left[ A_{j m}^{(m)} + i p A_{j m}^{(e)} \right],\] where \(j_0 = \max(|m|, 1)\), \(m = l + p\), \(C_j\) are beam coefficients, and \(A_{j m}^{(m/e)}\) are multipole fields computed via get_multipoles.

Visualizes the computed multipolar field sum. When called, the beam object is displayed.
Plots either individual polarization components (plot="components", showing \(\xi_1, \xi_0, \xi_{-1}\)) or total intensity (plot="total") across defined planes. The field intensity is computed as \(|E|^2\).