Discrete-Gauss states and the generation of focusing dark beams

Discrete-Gauss states are a new class of Gaussian solutions of the free Schrödinger equation owning discrete rotational symmetry. They are obtained by acting with a discrete deformation operator onto Laguerre-Gauss modes. We present a general analytical construction of these states and show the necessary and sufficient condition for them to host embedded dark beam structures. We unveil the intimate connection between discrete rotational symmetry, orbital angular momentum, and the generation of focusing dark beams. The distinguishing features of focusing dark beams are discussed. The potential applications of discrete-Gauss states in advanced optical trapping and quantum information processing are also briefly discussed.

Figure 1

Diffractive optical elements as mode converters: (a) a spiral phase plate transforms the fundamental $\left|\mathrm{LG}\right(0,0)\rangle$ mode into a higher-order $\left|\mathrm{LG}\right(3,0)\rangle$ mode with $q=+3$; (b) a DSDE with rotational order $N=4$ converts a $\left|\mathrm{LG}\right(3,0)\rangle$ mode into a multisingular ${|\mathrm{DG}(+3,0,4)\rangle }_v={|\mathrm{DG}(-1,1,4)\rangle }_v$ state (see Sec. 3 for the definition of the later notation in terms of $m$ and $k$). The amplitude of the field is represented as a yellow 3D surface (in this case, the one corresponding to 1/6 maximum). Trajectories of positively (negatively) charged singularities are represented in green (red).

Figure 2

Amplitude (3D surface at half-maximum) and singularity structure of a Laguerre-Gauss mode $\left|\mathrm{LG}\right(l,0)\rangle$ and two of its discrete deformations $\left|\mathrm{DG}\right(m,k,N)\rangle$ in the $xy\tau$ space [right column, top view; green (red) color indicates positive (negative) $q$; ${\tau }_R=1$; $v=0.1]$. (a) $\left|\mathrm{LG}\right(3,0)\rangle$ mode; single dark ray trajectory is shown. (b) DG state with $k=0$ and its single associated dark ray with $q_{\mathrm{ax}}=+3$. (c) DG state with $k=1$ generating a focusing dark beam with $q_{\mathrm{ax}}=-2$ and $N=5$ off-axis $q=+1$ singularities.

Figure 3

Evolution in $\tau$ of the amplitude [left column] and phase [right column] of a ${|\mathrm{DG}(-1,1,3)\rangle }_v$ state. Using the equivalent notation ${|\mathrm{DG}(l,N)}_v\rangle$ in terms of the “unfolded” discrete angular momentum $l$, this state is identical to the ${|\mathrm{DG}(+2,3)\rangle }_v$ state. The dashed white square in the $\tau =-3$ snapshot indicates the region that is enlarged for the representation of the phase.

Figure 4

Same as in Fig. 3 but for the ${|\mathrm{DG}(-1,1,3)\rangle }_v$ state, or ${|\mathrm{DG}(+4,3)\rangle }_v$ state, using equivalent ${|\mathrm{DG}(l,N)}_v\rangle$ notation.

Figure 5

Structure of focusing dark beams embedded in lower-order states near the dark focus for a quadruplet with $\left|k\right|=1$ and $\left|m\right|=1$ for $N=3$. The two states on the left column correspond to $l=\pm l_2=\pm 2$, whereas the ones in the right column correspond to $l=\pm l_1=\pm 4$. $[{\tau }_R=1$; $v=0.1$; ${\tau }_{\min }=-3$; ${\tau }_{\max }=3$; transverse range: left $L=0.75$ and right $L=1.5$.]

Figure 6

Dark focus structure of a DG state with $m=0$ and no axial dark ray $[{\tau }_R=1$; $v=0.25]$. (a) Amplitude at 1/4 maximum and dark beam with $q_{\text{ax}}=0$ and $q_{\text{df}}=-4$; 3D box is $8\times 8\times 20$. (b) Dark focus xz section. (c) 3D representation of the dark focus region (1/30 maximum); 3D box is $1\times \text{1}\times \text{2.5}$.

Figure 7

DG state with $k=2$ generating a focusing dark beam with a highly charged singularity on axis $[{\tau }_R=1$; $v=0.1]$. Phase profiles are represented at $\tau =-5$, 0, and $+5$. 3D box is $7\times 7\times 12$.