Shifted nondiffractive Bessel beams

Nondiffractive Bessel beams are well known to have infinite energy and infinite orbital angular momentum (OAM). However, when normalized to unity of energy, their OAM is finite. In this work, we derive an analytical relationship for calculating the normalized OAM of the superposition of off-axis Bessel beams characterized by the same topological charge. We show that if the constituent beams of the superposition have real-valued weight coefficients, the total OAM of the superposition of the Bessel beams equals that of an individual nonshifted Bessel beam. This property enables generating nondiffractive beams with different intensity distributions but identical OAM. The superposition of a set of identical Bessel beams centered on an arbitrary-radius circle is shown to be equivalent to an individual constituent Bessel beam put in the circle center. As a result of a complex shift of the Bessel beam, the transverse intensity distribution and OAM of the beam are also shown to change. We show that, in the superposition of two or more complex-shifted Bessel beams, the OAM may remain unchanged, while the intensity distribution is changed. Numerical simulation is in good agreement with theory.

Figure 1

(a) Intensity and (b) phase $(\pi$, black; −$\pi$, white) of the superposition of three shifted Bessel beams with parameters: $n=3,R_0=4\lambda ,\alpha =1/\lambda ,c=4,\gamma =\pi$, vector of weight coefficients $\mathbf{C}=[1,1,1]$. Frame size, $2R=60\lambda$.

Figure 2

(a) Encoded phase [Fig. 1] to generate a Bessel beam with the transverse intensity pattern shaped as an equilateral triangle and (b) experimentally generated intensity patterns at different distances from the plane $z=0$: (b) 0 mm, (c) 200 mm, and (d) 400 mm. Mesh step (distance between the ticks) is 0.5 mm.

Figure 3

Calculated (a) intensity and (b) phase $(\pi$, black; $-\pi$, white) patterns generated by the superposition of three shifted Bessel beams with parameters $n=5,R_0=8\lambda ,\alpha =1/\lambda ,c=3,\gamma =\pi$, weight coefficient vector, $\mathbf{C}=[1,1,1]$. Frame size, $2R=30\lambda$.

Figure 4

(a) Encoded phase to generate the superposition of three Bessel beams with the transverse intensity pattern featuring three bright spots [Fig. 3] and experimentally generated intensity patterns at different distances from the plane $z=0$: (b) 0 mm, (c) 200 mm, and (d) 400 mm. Mesh step (distance between the ticks), 0.5 mm.

Figure 5

(a) Intensity and (b) phase $(\pi$, black; $-\pi$, white color) of a superposition of four $\left(P=4\right)$ shifted Bessel beam with parameters $n=7,R_0=6\lambda ,\alpha =1/\lambda ,c=6,\gamma =\pi ,\mathbf{C}=[1,1,1,1]$. Frame size, $2R=60\lambda$.

Figure 6

SLM-aided (a) encoded phase of Fig. 5 to generate the superposition of four shifted Bessel beams with a square-shaped transverse intensity distribution and experimental intensity patterns at different distances from the plane $z=0$: (b) 0 mm, (c) 200 mm, and (d) 400 mm. Mesh step (distance between the ticks), 0.5 mm.

Figure 7

Simulated (a) intensity and (b) phase $(\pi$, black; -$\pi$, white) of the superposition of six shifted Bessel beams with parameters $P=6,n=10,R_0=12\lambda ,\alpha =1/\lambda ,c=12,\gamma =\pi ,\mathbf{C}=[1,1,1,1,1,1]$. Frame size, $2R=60\lambda$.

Figure 8

(a) SLM-aided encoded phase [Fig. 7] to generate the superposition of six shifted Bessel beams with a hexagon-shaped transverse intensity distribution and experimental intensity patterns at different distances from the plane $z=0$: (b) 0 mm, (c) 200 mm, and (d) 400 mm. Mesh step (distance between the ticks), 0.5 mm.

Figure 9

Transverse intensity distribution of the superposition of $P$ shifted Bessel beams at $P=5$ (a), 8 (b), 10 (c), 20 (d), 40 (e), and 60 (f). Frame size in all figures is $240\lambda \times 240\lambda$.

Figure 10

(a) SLM-aided encoded phase to generate the superposition of 40 shifted Bessel beams and experimental intensity patterns at different distances from the plane $z=0$: (b) 0 mm, (c) 50 mm, and (d) 100 mm. Mesh step (distance between the ticks), 0.5 mm.