Products of displaced Laguerre-Gaussian beams

TL;DR

The paper addresses how products of displaced Laguerre--Gaussian beams (pdLGBs) propagate in free space and how their transverse rotation encodes propagation information. It introduces an exact modal expansion based on the optical analogy of displaced number states to express each factor as a sum of origin-centered LGBs, yielding coefficients $C_{P,L}$. Propagation then becomes the independent evolution of weighted LGB modes with a common quadratic phase, radial scaling, and Gouy phases, and the net OAM in a centroid frame, $\,\langle \hat L_z \rangle$, governs rotation, revealing three regimes: no rotation for zero net OAM, rigid rotation for identical $p=0$ factors, and nonrigid rotation otherwise. This framework enables analytic control of transverse rotation for engineered structured light and has potential applications in depth-sensitive optical microscopy, imaging, and tracking.

Abstract

We study the free-space propagation of products of displaced Laguerre--Gaussian beams. Each displaced factor admits an exact representation as a superposition of standard Laguerre--Gaussian beam modes through the optical analog of displaced number states. We reduce the resulting product to a single modal expansion with closed-form weight coefficients and explicit azimuthal selection rules. Working in the reference frame defined by the centroid of the transverse displacements, we evaluate the net orbital angular momentum directly from the modal weights, which provides a criterion to predict transverse rotation. We identify three propagation regimes: no transverse rotation for zero net orbital angular momentum, rigid rotation for products of identical factors with zero radial index, and nonrigid rotation with intensity redistribution otherwise. Our framework enables engineering structured light beams whose transverse rotation encodes propagation information, relevant to depth-sensitive optical microscopy, imaging, and tracking.

Figures (7)

• Figure 1: (a) $\vert c_{u_j,v_j} \vert^{2}$ for a displaced LGB with $\{ p_j,\ell_j,(x_j,y_j) \} = \{0,1,(2w_{0},0) \}$ for $u_{j}, v_{j} \in \left\{ 0, 1, 2, \ldots, 10\right\}$. (b) Intensity and (c) phase distributions at $z \in \left\{ 0, 1, 2 \right\} z_{R}$ in the range $x, y \in [-5, 5] w_{0}$, with $I_{\mathrm{M}}$ being the maximum intensity value. Truncation artifacts are visible in the phase distribution at $z = 0$ and $z = z_R$.
• Figure 2: (a) Modal weights $\vert C_{P,L} \vert^{2}$ for the pdLGB formed by two displaced LGBs located at $\{ (x_{1},y_{1}), (x_{2},y_{2}) \} = \{(w_{0}/2, 0), (-w_{0}/2, 0)\}$ with opposite azimuthal charges, $\{ p_{1}, \ell_{1} \} = \{ 0, 1 \}$ and $\{ p_{2}, \ell_{2} \} = \{ 0, -1 \}$, yielding zero net OAM. (b) Intensity and (c) phase distributions at $z \in \{ 0, 1, 2 \} z_{R}$ in the range $x, y \in [ -5, 5 ] w_{0}$, with $I_{\mathrm{M}}$ the maximum intensity value. Truncation artifacts are visible in the phase distribution at $z=0$.
• Figure 3: Same as Fig. \ref{['fig:Fig2']} for two displaced LGBs with opposite azimuthal indices and identical higher radial order, $\{ p_{1}, \ell_{1} \} = \{ 1, 1 \}$ and $\{ p_{2}, \ell_{2} \} = \{ 1, -1 \}$, located at $\{ (x_{1},y_{1}), (x_{2},y_{2}) \} = \{(w_{0}, 0), (-w_{0}, 0)\}$.
• Figure 4: Same as Fig. \ref{['fig:Fig2']} for two identical factors with $\{ p_{1}, \ell_{1} \} = \{ p_{2}, \ell_{2} \} = \{ 0, 1 \}$.
• Figure 5: Same as Fig. \ref{['fig:Fig2']} for three identical factors with $\{ p_{1}, \ell_{1} \} = \{ p_{2}, \ell_{2} \} = \{ p_{3}, \ell_{3} \} = \{ 0, 1 \}$ located at $\{ (x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3}) \} = \{ (w_{0}/2,0), (-w_{0}/4,\sqrt{3}w_{0}/4), (-w_{0}/4,-\sqrt{3}w_{0}/4) \}$.