# Simultaneous Generation of Complex Structured Curve Beam

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{0}(t), y

_{0}(t), z

_{0}(t)] represents the prescribed curve in the Cartesian coordinate with t∈[0,2π]. f

_{0}and λ refer to the focal length of the Fourier lens and the wavelength, respectively.

_{0}(t) = Rcos(t) y

_{0}(t) = Rsin(t) and demonstrate the performance of this technique by simulation. The intensity distribution of the resulting beam is displayed in Figure 1b. The phase distribution of the ring is well defined along curves under the topological charge of m = 1 (see Figure 1f). Besides this, we consider other three shapes: an Archimedean spiral, a trefoil-knotted curve and a square curve. The topological charge is m = 1 for all the curves. For the sake of clarity, the corresponding curve parametric expressions are provided in Table 1. The intensity distribution and the phase distribution of the resulting beam is displayed in Figure 1b–i.

_{i}and v

_{i}are the spatial coordinates of the generated beam in the far field, achieved with a Fourier lens of focal length f

_{0}. K = 2π/λ is the wave number. z

_{i}is the axial shifted displacement away from the focal plane (Fourier plane). In order to generate curve beams simultaneously, the expressions of the final complex CGH need to be added together by

_{i}, u

_{i}, v

_{i}and m) in the CGH calculation.

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Computer-generated hologram (CGH) of a 2D ring curve beam (

**b**) Scheme of holographic three-dimensional beam shaping technique. (

**c**,

**g**) Reconstructed intensity and phase distribution of the ring curve at the focal plane. (

**d**,

**h**) Reconstructed intensity and phase distribution of the Archimedean spiral at the focal plane. (

**e**,

**i**) Reconstructed intensity and phase distribution of the trefoil-knotted curve at the focal plane. (

**f**,

**j**) Reconstructed intensity and phase distribution of the square curve at the focal plane.

**Figure 2.**(

**a**,

**c**) Reconstructed intensity of the multiple light beams at the focal plane. (

**b**,

**d**) Phase distribution of the multiple light beams at the focal plane.

**Figure 3.**The beams focused on different regions. A three-dimensional schematic of focused beams is seen in (

**a**). Intensity distribution of the beams projected in the focal plane (z = 0) seen in (

**c**), and the Archimedean spiral and the trefoil-knotted curve are focused on the z = −0.05 m and z = 0.05 m planes, respectively, seen in (

**b**,

**d**). (

**e**–

**g**) The phase distribution of the multiple light beams.

**Figure 4.**The beams focused on different regions. A three-dimensional schematic of focused beams is seen in (

**a**). Intensity distribution of the beams projected in the focal plane (z = 0) seen in (

**c**), and the Archimedean spiral and the trefoil-knotted curve are focused on the z = −0.05 m and z = 0.05 m planes, respectively, seen in (

**b**,

**d**). (

**e**–

**g**) The phase distribution of the multiple light beams.

**Figure 5.**Experimental setup. The hologram is addressed into the SLM, which is illuminated by a collimated laser beam. After the beam passes through lens 1, the desired pattern can be filtered with a diaphragm. Then resulted beams pass through lens 2 and lens 3, and can be captured by the camera.

**Figure 6.**(

**a**,

**b**) Experimental results. The resulting beams are photographed on the focal plane. (

**c**–

**j**) Experimental results of the four different curve beams in two different types of three-dimensional layouts, under different analyzer directions.

Type of Curve | x_{0}(t) | y_{0}(t) |
---|---|---|

ring curve | Rcos(t) | Rsin(t) |

Archimedean spiral | −Rtcos(10t) | −Rtsin(10t) |

trefoil-knotted curve | Rcos(t) − 2Rcos(2t) | Rsin(t) + 2Rsin(2t) |

square curve | −2Rcos(t) + 0.3Rcos(kt) | −2Rsin(t) + 0.3Rsin(kt) |

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**MDPI and ACS Style**

Wu, J.; Tang, X.; Xia, J.
Simultaneous Generation of Complex Structured Curve Beam. *Nanomaterials* **2019**, *9*, 87.
https://doi.org/10.3390/nano9010087

**AMA Style**

Wu J, Tang X, Xia J.
Simultaneous Generation of Complex Structured Curve Beam. *Nanomaterials*. 2019; 9(1):87.
https://doi.org/10.3390/nano9010087

**Chicago/Turabian Style**

Wu, Jun, Xinquan Tang, and Jun Xia.
2019. "Simultaneous Generation of Complex Structured Curve Beam" *Nanomaterials* 9, no. 1: 87.
https://doi.org/10.3390/nano9010087