# Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation

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## Abstract

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## Featured Application

**Controlling the spectral Gouy rotation of propagating pulsed beams with orbital angular momentum opens specific channels for ultrafast communication and excitation.**

## Abstract

## 1. Introduction

## 2. Theoretical Background and Numerical Simulation

#### 2.1. Spectral Gouy Rotation of Bessel–Gauss Beams

_{B}of a Bessel beam compared to a reference plane wave accumulates with the distance z [50]:

#### 2.2. Numerical Simulation

_{0}= 100 µm to emulate the diffraction at components with small apertures. To filter out essential features of the obtained spatio-spectral maps, local statistical moments M

^{(i)}(x,y,λ) were determined. At the same time, this kind of spectral image processing procedure also enables one to reduce the large data containing fully extended spectra for each point in space. Figure 1 shows the results of the numerical simulations in terms of centers of gravity (COG), i.e., first spectral moments M

^{(1)}(x,y) for selected values of z between 4 and 11 mm increasing in steps of ∆z = 0.5 mm (see also the visualization movie in the supplementary material). The color code of the graphics reaches from about 793 nm (blue-shifted parts) until about 808 nm (red-shifted parts). The limits of the spectral interval slightly change depending on the axial distance to best visualize the spectral maps. The variation of the applied scale, however, did not exceed a value of 0.9 nm. The divergent Gaussian beam causes a stretching of the spatial scale with increasing distance.

## 3. Experimental Techniques and Mathematical Tools

#### 3.1. Shaping of Femtosecond Vortex Pulses with Stationary and Adaptive Components

#### 3.2. Vortex Analysis with Spectral Statistical Moments

^{(j)}(x,y) and linear combinations of such were determined for each transversal coordinate (x,y) in a plane. The resulting “movies” of z-dependent moment maps M

^{(j)}(x,y,z) specifically describe the spectral propagation behavior. The index j indicates the order of the statistical moment. The moments for j = 1 to j = 4 are related to center of gravity (COG), standard deviation, skewness, and kurtosis. More details on the statistical methods, including an extension to global moment analysis with radial meta-moments, were reported in ref. [55].

#### 3.3. Rotation Control and Relevant Parameters

_{z}can be defined as the axial derivative of the rotation angle φ of the connecting line between these COG coordinates:

_{t}follows as

_{0}and T

_{0}= P

_{0}/c within a given propagation distance z is directly proportional to Ω

_{z}and Ω

_{t}, respectively. The rotation angle as a function of the distance can be written as

## 4. Experimental Results and Discussion

#### 4.1. Parameters of Applied Spiral Phase Shapers

#### 4.2. Spectral Control by Tuning Center Wavelength and Bandwidth

#### 4.3. Variation of Grating Parameters

_{z}of the spectral Gouy rotation was obtained by (i) subsequently inserting static SPGs with different structural periods, and (ii) by adaptively modifying the periods of spiral phase gratings programmed into grey value maps of an LCoS-SLM.

_{max}~ arctan(λ/2D). For an element diameter of 200 µm and a wavelength of 800 nm, θ

_{max}should be in the range of 0.115°.

_{j}results from different starting points z

_{j}of the zones of detectable rotation. The accumulated phase is found to be linear with small deviations, as is expected for Bessel-like beams. Linearity enables well-defined rotation control. For some curves, the ranges of parameters are not fully covered because some of the measurements had to be terminated for technical reasons like laser stability.

#### 4.4. Rotation Control by Chirped Spirals and Multibeam Superposition

_{1}(t) and Ω

_{2}(t), we can define a spectral self-torque

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Numerical simulation of rotating anomalies in the spatio-spectral propagation of femtosecond vortex pulses in air (FOV = 12 × 12 µm

^{2}). The centers of gravity (COG) of local spectral moments at distances z between 4 and 11 mm show the circulation of spectral eyes around the singularity during a full period of spectral Gouy rotation. To induce the orbital angular momentum, a binary spiral phase grating with a period of 8 µm and a phase stroke of 2π at a central wavelength of 800 nm was illuminated by a Gaussian beam of a full-width-at-half-maximum (FWHM) spectral bandwidth of 12 nm with a waist radius of w

_{0}= 100 µm. The color code reaches from a lower limit (blue tail) of λ

_{min}= 793 nm to an upper limit (red tail) of λ

_{max}= 808 nm (±0.9 nm).

**Figure 2.**Experimental setup for the generation and detection of femtosecond vortex pulses. An orbital angular momentum is induced by optional use of a spatial light modulator (SLM), a transmissive diffractive spiral phase grating (SPG), or a reflective spiral phase plate (SPP) as beam shaper. The vortex is magnified by a microscope and analyzed in spectral domain by a scanning fiber-based spectrometer. The fiber is moved by high-precision xyz-translation stages. From detected local spectra, 2D maps of spectral statistical moments M

^{(j)}(x,y) are extracted for each distance z.

**Figure 3.**Parts of selected spiral phase gratings (SPGs) used as vortex beam shapers in the experimental studies: (

**a**–

**c**) fixed diffractive elements (left) and (

**d**) spiral structure programmed into the SLM grey scale map (right). The elements correspond to configurations 3, 4, 6, and 2 in Table 1, respectively.

**Figure 4.**Spectral control by (

**a**) tuning the center wavelength via laser mode adjustment (1–3: three states shown with center wavelengths at 798 nm, 800 nm, and 802 nm, respectively), (

**b**) varying the FWHM spectral bandwidth ∆λ for a single selected laser mode (center wavelength λ

_{0}= 798 nm) between 2.5 nm and 8.5 nm.

**Figure 5.**Spectral transfer into vortex beams as a function of input spectral FWHM bandwidth for three wavelength ranges around (1) 798 nm, (2) 800 nm, and (3) 802 nm (spectral profiles in Figure 4a). Filled symbols: upper limits, hollow symbols: lower limits of spectral content in the spectral eyes.

**Figure 6.**Propagation-dependent maps of 1st spectral moment (COG) of experimentally detected maps of local spectra for two different vortex shaper configurations programmed in an LCoS-SLM (aperture D = 2 mm): (

**a**) generation of a Bessel–Gauss beam with a spiral phase grating (SPG) with spectral Gouy rotation (spiral grating period p = 16 µm) (see also video S1 in the supplementary material); (

**b**) vortex generation with a spiral phase plate (SPP), without spectral Gouy rotation. The little grey scale pictures at the left side symbolize the type of element (SPG, SPP).

**Figure 7.**Controlling the angular velocity by varying geometrical parameters. The modulo of the rotation angle is plotted as a function of propagation distance for equal paths ∆z. Different grating structure periods p and overall diameters D for programmable (SLM) and fixed (DOE) configurations are compared: (

**a**) SLM, p = 32 µm, D = 2 mm; (

**b**) SLM, p = 16 µm, D = 2 mm; (

**c**) DOE, p = 32 µm, D = 0.4 mm; (

**d**) DOE, p = 16 µm, D = 2 mm; (

**e**) DOE, p = 16 µm, D = 0.2 mm; and (

**f**) DOE, p = 16 µm, D = 0.4 mm.

**Figure 8.**Axial rotation periods P

_{0}of the spectral eyes as a function of conical beam angle θ and element diameter D (DOE = fixed spiral grating, SLM = programmable spatial light modulator). It is indicated that both parameters can be used to control the spectral Gouy rotation.

**Figure 9.**Angle of spectral Gouy rotation for extended propagation distances (numbers correspond to configuration numbers in Table 1). The increase of phase appears to be nearly linear.

**Table 1.**Geometrical parameters of spiral phase gratings used as orbital angular momentum (OAM) beam shapers.

Spiral Phase Gratings (SPG) Con-Figuration | Type of Orbital Angular Momentum (OAM) Shaper | Grating Period (µm) | Aperture Diameter (mm) | Depth of Focal Zone (mm) | Fresnel Number (Aperture)^{1} | 1st Order Diffraction Angle (°) |
---|---|---|---|---|---|---|

1 | SLM_32 | 32 | 2 | 40 | 250 | 1.43 |

2 | SLM_16 | 16 | 2 | 20 | 500 | 2.87 |

3 | DOE 4-step | 16 | 2 | 20 | 500 | 2.87 |

4 | DOE 2-step | 16 | 0.4 | 4 | 100 | 2.87 |

5 | DOE 2-step | 16 | 0.2 | 2 | 50 | 2.87 |

6 | DOE 2-step | 32 | 0.4 | 8 | 50 | 1.43 |

^{1}Data related to half depth of each focal zone at a central wavelength of 800 nm.

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## Share and Cite

**MDPI and ACS Style**

Liebmann, M.; Treffer, A.; Bock, M.; Wallrabe, U.; Grunwald, R. Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation. *Appl. Sci.* **2020**, *10*, 4288.
https://doi.org/10.3390/app10124288

**AMA Style**

Liebmann M, Treffer A, Bock M, Wallrabe U, Grunwald R. Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation. *Applied Sciences*. 2020; 10(12):4288.
https://doi.org/10.3390/app10124288

**Chicago/Turabian Style**

Liebmann, Max, Alexander Treffer, Martin Bock, Ulrike Wallrabe, and Ruediger Grunwald. 2020. "Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation" *Applied Sciences* 10, no. 12: 4288.
https://doi.org/10.3390/app10124288