Operator based propagation of Whittaker and Helmholtz Gauss beams

Abstract

We introduce a compact operator-based technique that solves the paraxial wave equation for a broad class of structured light fields. Using the spatial evolution operator to propagate two families of physically apodized inputs, Gaussian apodized Whittaker integrals and Gaussian apodized Helmholtz fields, we derive closed form expressions that retain the Gaussian width and therefore describe finite energy beams. The method unifies and extends the Helmholtz Gauss families and readily generalizes to nonseparable nondiffracting architectures. Experiments on superposed Bessel Gauss beams confirm the predicted transverse rotations, demonstrating that this operator approach is a fast, transparent, and practical alternative to standard diffraction ntegral treatments

Figures (3)

• Figure 1: Intensity distributions for the superposition of Bessel-Gauss beams which are characterized by $(m_1 = 1, \, {k_\perp}_1 = 1800 \, \mathrm{m}^{-1})$, and $(m_2 = -1, \, {k_\perp}_2 = 8900 \, \mathrm{m}^{-1})$, at three transverse planes: ($\mathrm{a}_1$) at $z = 0.00 \, \mathrm{m}$, ($\mathrm{b}_1$) at $z = 0.25 \, \mathrm{m}$ and ($\mathrm{c}_1$) at $z = 0.75 \, \mathrm{m}$. The corresponding experimental distributions are shown in ($\mathrm{a}_2$)-($\mathrm{c}_2$). The experimental parameters are $g = 0.25 \times 10^{7} \, \mathrm{m}^{-2}$ and $\lambda = 632.8 \, \mathrm{nm}$, all within a viewing window of $4 \, \mathrm{mm}$.
• Figure 2: Intensity distributions for the superposition of Bessel-Gauss beams which are characterized by $(m_1 = 1, \, {k_\perp}_1 = 1800 \, \mathrm{m}^{-1})$, and $(m_2 = 0, \, {k_\perp}_2 = 8900 \, \mathrm{m}^{-1})$, at the same planes as in Fig. \ref{['figura1']}. The corresponding experimental distributions are shown in ($\mathrm{a}_2$)-($\mathrm{c}_2$). The parameters of $g$ and $\lambda$, as well as the observation window width are the same as those in the Fig. \ref{['figura1']}.
• Figure 3: Intensity distributions for the superposition of Bessel-Gauss beams which are determined by $(m_1 = 1, \, {k_\perp}_1 = 1800 \, \mathrm{m}^{-1})$, and $(m_2 = 1, \, {k_\perp}_2 = 8900 \, \mathrm{m}^{-1})$, at the same planes as in Fig. \ref{['figura1']}. The corresponding experimental distributions are shown in ($\mathrm{a}_2$)-($\mathrm{c}_2$). The parameters of $g$ and $\lambda$, as well as the observation window width are the same as those in the Fig. \ref{['figura1']}.