Mitosis-like dynamic for conservation of OAM

Abstract

Optical states characterized by an electromagnetic field with a spiral azimuthal phase are known as orbital angular momentum (OAM) states. Examples of structured light carrying OAM states include Laguerre-Gauss (LG) beams, which can be transformed into Hermite-Gauss (HG) beams through astigmatic transformations using cylindrical or spherical lenses. In this study, we explore the dynamics of OAM conservation by combining two noncommutative operations: astigmatic transformation, achieved with a tilted spherical lens, and an up-conversion process facilitated by a ${\chi }^{\left(2\right)}$ crystal. In this experiment, we observe that the intensity distribution in the second harmonic (SH) beam undergoes a separation process resembling the stages of mitosis, a biological process where a single cell divides into two genetically identical daughter cells. The transformations in the SH beam closely mirror key mitotic phases, including metaphase, anaphase, and telophase. To validate these similarities, we analyze the intensity, phase, and electric field profiles obtained from numerical simulations. These findings are further supported by experimental results and enhanced through phase retrieval techniques.

Figure 1

Schematic of astigmatic transformation of LG beams and their up-conversion process. In normal incidence, an LG$_{3,0}^{}$ beam is focused and frequency doubled through a second order nonlinear (${\chi }^{(2)}$) medium into an LG$_{6,0}^{}$ beam with twice the amount of OAM per photon. By tilting the focusing spherical lens along $x-$ or $y-$axis, different pump profiles are generated. We image the incident, the astigmatically transformed, and the converted beam profiles on several critical positions along the propagation direction. These positions are: (i) structured light generated by a spatial light modulator (SLM), (ii) fundamental beam transformed astigmatically with a tilted spherical lens, (iii) SH signal right after the ${\chi }^{(2)}$ crystal, and (iv) SH signal at the far-field.

Figure 2

Numerical simulation of astigmatically transformed followed by SH process of LG$_{3,0}^{}$. (a) Intensity and phase profiles of the fundamental [(i), (ii)] and the SH signal [(iii), (iv)] for $\alpha =0$, ${10}^{\circ }$, and ${15}^{\circ }$. (b) overlay of SHG electric field amplitude and phase distribution for $\alpha \in [0^{\circ },{15}^{\circ }]$ to show mutation of the singularity into a mitosis-like process visible in the far-field on the plane (iv). The singularities are marked with white dotted circle.

Figure 3

Dynamic of orbital angular momentum conservation. Numerical results of upconverted gHLG$_{3,1}^{\alpha }$ in plane Fig. 1(iv). The SH signal intensity profile, phase profile, and overlay of electric field amplitude and phase profiles are generated with a fundamental pump obtained through astigmatic transform of LG$_{3,1}^{}$ beam under different tilted angle $\alpha \in [0,{15}^{\circ }]$ of the spherical lens. On the electric field images, the white dash-line circles point out the vortices while the white crosses show the emergence of anti-vortices.

Figure 4

Experimental results of LG$_{3,0}^{}$ beam during astigmatic transformation and SHG process. Intensity profiles are taken at the planes [(i), (iv)] in Fig. 1. (a) Intensity profiles of the fundamental [(i), (ii)] and the SH signal [(iii), (iv)] for $\alpha =0$, ${10}^{\circ }$, and ${15}^{\circ }$. Phase profiles are also shown for signal on the plane (i) and (iv). (b) Experimental SHG electric field distribution for $\alpha \in [0^{\circ },{15}^{\circ }]$ in the far-field on the plane (iv). The singularities are marked with white dotted circle.

Figure 5

Analysis of singularity’s diameter between gHLG$_{\ell ,p}^{15}$ and its SHG signal. (a) Comparison in line profiles between gHLG$_{3,0}^{15}$ and its SHG signal in numerical model. (b) Comparison in line profiles between gHLG$_{3,0}^{15}$ and its SHG signal experimentally. (c) Ratio of singularity diameter of intensity at plane (iii) and plane (ii) as a function of topological charge of the input beam at plane (i).

Figure 6

Experimental results of SHG signal generated with LG$_{31}^{}$ beam astigmatically transformed with various tilting angles $\alpha \in$ [$0^{\circ },{15}^{\circ }$] to serve as the fundamental input. The first row correspond to the intensity profile collected at plane (iii). From the second and to the fourth row are presented intensity, phase profile and electric field distribution of the upconverted beam captured at plane (iv).

Figure 7

Experimental setup schematic. A polarization controlled femtosecond laser beam, centered at 808 nm with 3 kHz repetition rate, 38 fs pulse width, and 1.67 mJ pulse energy is incident on a spatial light modulator (SLM). The polarization control is achieved using a polarizer (P) and a half-wave plate (HWP). An iris is used for mode selection to isolate any desired LG$_{\ell ,p}^{}$ before it is propagated to plane (i). A tilted lens (TL), with $\alpha$ being the tilt angle, transforms the LG$_{\ell ,p}^{}$ modes into generalized HLG (gHLG$_{\ell ,p}^{\alpha }$) mode. A beam splitter (BS, not illustrated) is placed after the TL to image the intensity profile at the plane (ii). The upconverted beam through a $\beta$–Barium Borate crystal (BBO) is then separated from the fundamental beam using a bandpass filter (405 ± 10 nm). The SH beam is inverse Fourier transformed by lens L$_1^{}$ and imaged by Detector 1, plane (iv). Lenses L$_2^{}$, together with L$_1^{}$ are used to transfer plane (iii) onto Detector 2.