# Shack-Hartmann Wavefront Sensing of Ultrashort Optical Vortices

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Methods for Wavefront Characterization of Ultrashort Vortex Beams

#### Optical Vortex Detection and Wavefront Reconstruction

## 3. Results

#### 3.1. Intensity and Wavefront Characterization of Ultrashort Vortex Beams

#### 3.2. Modal Purity of the Vortex Beams

#### 3.3. Effect of Radial Aperturing on Modal Content

#### 3.4. Effect of Spiral Phase Plate Displacement on the Modal Composition

#### 3.5. Topological Charge Measurement at 10 Hz

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Vortex beam wavefront characterization setup. (

**a**) Experimental setup for characterization of IR vortex beams. A low-energy ( ∼0.1 nJ) IR beam is passed through a spiral phase plate (SPP) to generate a vortex beam. The resulting beam is loosely focused by a 2 m focal length lens and directed towards a Shack-Hartmann wavefront sensor (HASO) located ∼300 mm after the focal plane. To generate a higher topological charge vortex, multiple SPPs of $\ell =1$ are inserted in the beam. In the insets (

**b**,

**c**), we show the working principle of the Shack-Hartmann wavefront sensor. The incoming beam is sampled by the microlens array of the sensor. Depending on the local wavevector direction, the sampled beamlets are displaced concerning the reference spots of the calibrated wavefront sensor, as shown in (

**c**). The sampled spot distribution is acquired on a CCD camera located at a given distance (L) from the microlens array. An example raw Hartmanngram is presented in (

**d**) for a vortex beam of $\ell =4$. The annular intensity profile of the vortex beam is evident from the central dark sub-pupils. From the raw Hartmanngram, the local slope, and ultimately the wavefront can be retrieved. In (

**e**,

**f**), the map of the measured slope for $\ell =+4$ and $\ell =-4$ are shown. The slopes maps show a spiraling behavior around the center, whose direction of rotation changes from clockwise (

**e**) to counterclockwise (

**f**) with a change in sign of ℓ.

**Figure 2.**Characterization of low topological charge vortex beam. Reconstructed intensity (

**left**) and wavefront (

**right**) from single-shot raw Hartmanngram for beam of $\ell =+1$ (

**a**) and $\ell =-1$ (

**c**). The intensity distribution exhibits a dark central spot in both cases. The vertical axis in (

**b**,

**d**) represents the wavefront in the unit of wavelength $\lambda $. The wavefronts manifest a helical structure with an overall peak-to-valley value of ∼1$\lambda $, designating the unit topological charge. The handedness of wavefront rotation changes from clockwise (

**b**) to counterclockwise (

**d**) when the sign of the topological charge changes from positive to negative.

**Figure 3.**Characterization of vortex beams of higher topological charge. Intensity (

**top**) and wavefront (

**bottom**) for $\ell =$ 2 (

**a**), 3 (

**c**), and 4 (

**e**). In all the cases, the intensity distribution presents an annular profile, whereas the wavefronts show a helical structure. The wavefront twists by ∼2.005$\lambda $, ∼3.003$\lambda $, and ∼4.0$\lambda $ for $\ell =$ 2 (

**b**), 3 (

**d**), and 4 (

**f**), respectively. The clockwise wavefront rotation indicates the positive sign of the topological charge.

**Figure 4.**Fidelity of characterization: comparison between backpropagated and experimental intensity profile of vortex beams at the waist. We use the complex field reconstructed by the wavefront sensor to obtain intensity distribution (

**top**row) and the wavefront (

**middle**row) at the waist for $\ell =+1$ (

**a**,

**d**,

**g**), $+2$ (

**b**,

**e**,

**h**), and $-4$ (

**c**,

**f**,

**i**). The backpropagated wavefronts present a variation of 1, 2, and 4$\lambda $ for $|\ell |=1$, 2, and 4, respectively. For each case, the bottom row represents the experimentally obtained intensity profile at the waist. For all the configurations, the backpropagated and experimental profiles show an excellent agreement, indicating that the characterized amplitude and phase by the wavefront sensor indeed corresponds to the generated vortex beams.

**Figure 5.**Modal analysis of the vortex beams. The $\ell ,p$ modal content for (

**a**–

**d**). The global LG spectrum is normalized to unity. The contribution to the desired OAM mode is (

**a**) ∼94%, (

**b**,

**c**) ∼93%, and (

**d**) ∼92%, indicating high OAM purity in all the cases. The energy fraction contained in the dominant azimuthal and principal ($p=$ 0) radial mode is: (

**a**) ∼86%, (

**b**) ∼79%, (

**c**) ∼73%, and (

**d**) ∼66%.

**Figure 6.**Modal analysis for the vortex beams of topological charge $\ell =$$-5$ to $+5$. (

**a**) Comparison of the OAM purity obtained via azimuthal Fourier transform and LG decomposition. (

**b**) The energy contained in $p=$ 0 radial mode summed over $\ell =$$-10$ to $\ell =$$+10$. (

**c**) Contribution to $L{G}_{\ell =\ell ,\phantom{\rule{0.222222em}{0ex}}p=0}$ mode. (

**d**) The standard deviation of p-spectrum ($\Delta p$). The increasing trend of ($\Delta p$) with topological charge signifies a broader p-spectrum.

**Figure 7.**Effect of radial clipping on the modal content of $\ell =-3$ vortex beam. (

**a**) Intensity, (

**b**) reconstructed wavefront in the unit of wavelength, and (

**c**) modal content for unclipped beam. In (

**d**–

**f**), corresponding intensity, wavefront, and modal composition are shown for ∼16 mm iris diameter. The PtV wavefront signifying the topological charge is ∼3$\lambda $ in both cases. The OAM purity, i.e., contribution to $\ell =$$-3$ mode is ∼91%, ∼89% for fully open and 16 mm iris diameter, respectively. The contribution to $L{G}_{-3,\phantom{\rule{0.222222em}{0ex}}0}$ mode plunges from ∼69% to ∼45% for the latter case.

**Figure 8.**Effect of radial clipping on the modal content. The OAM purity and $L{G}_{\ell ,\phantom{\rule{0.222222em}{0ex}}0}$ mode contribution for various iris diameters: (

**a**) $\ell =\pm 1$, (

**b**) $\ell =\pm 2$, (

**c**) $\ell =\pm 3$, and (

**d**) $\ell =\pm 4$. The 30 mm aperture diameter corresponds to a fully open iris. The contribution to desired OAM order only minutely varies with aperture size, whereas the energy contained in $p=$ 0 radial mode shows a large reduction for the smaller diameter. This behavior is consistent for all the cases.

**Figure 9.**Effect of SPP displacement on the modal composition of $\ell =+1$ vortex beam. Intensity (left) and OAM spectra (center) for SPP displacement $\alpha ={r}_{0}/\omega =0.2$ (

**a**,

**b**) and $0.5$ (

**c**,

**d**). In (

**e**), a comparison between theoretical (solid line) and experimental (cross symbol) average OAM for various SPP shifts is presented. The theoretical and experimental results are in very good agreement and indicate the fractional total OAM of asymmetric Gaussian vortices.

**Figure 10.**Wavefront twist measurement of femtosecond OV at 10 Hz repetition rate. Realtime PtV wavefront measurement over 2000 laser shots for (

**a**) $\ell =$ 1, (

**b**) $\ell =$ 2, (

**c**) $\ell =$ 3, and (

**d**) $\ell =$ 4. The mean PtV wavefront value and standard deviation (SD) of the measurement are included in the inset of each plot. In all the cases, the mean PtV wavefront indicating topological charge is within 0.5% of the theoretically expected value.

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Pandey, A.K.; Larrieu, T.; Dovillaire, G.; Kazamias, S.; Guilbaud, O.
Shack-Hartmann Wavefront Sensing of Ultrashort Optical Vortices. *Sensors* **2022**, *22*, 132.
https://doi.org/10.3390/s22010132

**AMA Style**

Pandey AK, Larrieu T, Dovillaire G, Kazamias S, Guilbaud O.
Shack-Hartmann Wavefront Sensing of Ultrashort Optical Vortices. *Sensors*. 2022; 22(1):132.
https://doi.org/10.3390/s22010132

**Chicago/Turabian Style**

Pandey, Alok Kumar, Tanguy Larrieu, Guillaume Dovillaire, Sophie Kazamias, and Olivier Guilbaud.
2022. "Shack-Hartmann Wavefront Sensing of Ultrashort Optical Vortices" *Sensors* 22, no. 1: 132.
https://doi.org/10.3390/s22010132