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 $\chi ^{(2)}$ 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.

Introduction

Light beams carrying orbital angular momentum (OAM) are purposefully and carefully designed such that their Poynting vector is spatially varying as they propagate. This property can lead to new phenomena borne out of light-matter interaction that can expand the functionality and enhance the capability of optical systems.

The most common examples of the vortex beam are the Laguerre-Gaussian (LG) modes^1,2,3,4,5, which have a phase singularity (optical vortex) along their propagation axis. A LG beam is identified by its azimuthal and radial functions. The azimuthal function is described by a helical phase $\exp (i\ell \phi )$ where $\phi$ is the azimuthal angle and $\ell$ the so-called topological charge (TC) corresponding to an orbital angular momentum of $\ell \hbar$ per photon⁶.

The use of liquid crystal based spatial light modulator (SLM) has introduced flexibility in the generation and manipulation of structured light beams^7,8,9,10. Such flexibility has enabled numerous applications including, but not limited to topological charge transfer in light-matter interactions, atomic transitions, spin object detection, intrinsic OAM generating laser, spin-orbital coupling, and quantum information and communication^{11,12,13,14,15,16,17,18,19,20,21,22}. In quantum communication, there is a need for quantum mode and frequency conversion devices^{23,24,25,26,27,28,29,30,31,32}, e.g., based on three or four wave mixing³³. Furthermore, the application of astigmatic transformations for the interchange of LG and Hermite-Gaussian (HG) beams^34,35,36,37 is well established.

Recently, research has been conducted to show that astigmatic transformation and nonlinear processes such as second harmonic generation (SHG) are not commutative operations^38,39. As a consequence, the reversal in the order of these two operations results in different beam profiles. This discovery furthered new avenues for the exploration of novel light formation by combining different transformation processes. To date, we know of no experimental studies on the mechanism of OAM conservation in optical beams undergoing an astigmatic transformation coupled with the up-conversion process.

Here, we present a thorough investigation into the dynamic of the second harmonic (SH) process initiated by an astigmatically transformed LG beam that generates a generalized Hermite-Laguerre-Gaussian (gHLG) mode⁴⁰ whereby the momentum is conserved for the whole process. By increasing the tilt angle of the focusing lens ($\alpha \in [0,15^\circ ]$), we generate different gHLG$^\alpha$ beam profiles incident into the $\beta -$Barium Borate (BBO) crystal, enabling the generation of new intensity profiles and the study of the angular momentum dynamics. Furthermore, we observe a “cellular-like” division ⁴¹ in the spatial distribution of singularities in the azimuthal function, driven by the astigmatic transformation. Additionally, the radial function of vortex beams breaks into a pair of singularities. Our finding includes both experimental data and numerical modeling of the beam evolution at several critical positions, including the SH signal immediately after the nonlinear medium.

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.

Generally, our experimental electric field reconstruction obtained from the phase retrieval method agrees well with that of our numerical model.

Numerical model

The HG and LG modes are solutions to the Helmholtz equation and can be represented on a higher-order OAM Poincaré sphere ^42,43. It has been shown that an LG$_{\ell ,p}$ mode can be decomposed into a superposition of HG$_{m,n}$ modes. Likewise, a diagonal HG beam can be decomposed into a similar set of HG modes^34,37. However, these two decompositions differ by a phase of ${\pi }/{2}$ between consecutive modes. That phase shift allows their interconversion through astigmatic transformations, which manipulate the mode-dependent Gouy phase, with a tilted spherical lens ^44,45 or a pair of cylindrical and spherical lenses ^34,35,36,37. Abramochkin et al.⁴⁰ presented an analytic expression of the gHLG beam which describes the transformation between HG and LG beams. To study the creation of a new intensity profile and its OAM conservation mechanism owing to the non-commutativity ³⁹ between the astigmatic transform and the up-conversion, instead of using the gHLG analytic expression, we performed numerical simulations of the beam evolution along its propagation path and compared to experimental beam images sampled on four critical planes.

The first plane−We start by considering an LG mode profile, as illustrated in Fig. 1 plane (i). The general electric field $u^{LG}_{\ell ,p} (r,\varphi , z')$ of an LG$_{\ell ,p}$ beam is expressed in cylindrical coordinates ($r,\varphi , z'$) as shown in Eq. (1).

$$\begin{aligned} \begin{aligned} u_{\ell ,p}^{\text {LG}}(r,\varphi ,z')= A(r,z')e^{i\ell \varphi } \\ A(r,z')= {\frac{1}{w(z')}} {\sqrt{\frac{2}{\pi }\frac{p!}{(\left| {\ell }\right| +p)!}}}~{\left[ \frac{r\sqrt{2}}{w(z')} \right] }^{\left| {\ell }\right| }~{\mathcal {L}_p^{\left| {\ell }\right| }\left( \frac{2r^2}{w^2(z')}\right) }~{\exp (-\frac{r^2}{w^2(z')})}~{\exp (\frac{iz'kr^2}{2({z'}^2+z^2_R)})}~{\exp [-i(2p + \left| {\ell }\right| + 1)\phi (z')]} \end{aligned} \end{aligned}$$

(1)

Here, k is the wave-number, $\mathcal {L}_p^{|\ell |}(r)$ is the generalized Laguerre polynomial depending on the radial function index p and the topological charge $\ell$–both are integers, $z_R = k w_0^2/2$ is the Rayleigh range, $w(z')$ is the beam radius defined as $w(z')= w_o[{1 + ({z'}/{z_R})^2}]^{1/2}$ with $w_0$ being the waist radius at $z'=0$, the LG’s Gouy phase is defined as $\exp [-i(2p+\left| {\ell }\right| +1)\phi (z')]$ with $\phi (z')=\tan ^{-1}({z'}/{z_R})$ being the Gouy phase of a Gaussian mode. Here, we take $u_{\ell ,p}^{\text {LG}}(r,\varphi ,z'=0)$ as our input beam and choose the laboratory axis $z=0$ that starts with the plane (i).

The second plane−Before reaching the plane (ii), we apply two operators to the electric field of LG$_{\ell ,p}$ in Eq. (1). The first is the tilted lens phase $\Phi _\alpha$ and the second is the propagator noted $\mathcal {P}(\Delta z)$. A spherical lens tilted by an angle $\alpha$ with respect to $x-$axis ($y-$axis) performs an astigmatic transformation of the incident LG$_{\ell , p}$ mode into a gHLG mode. Astigmatic conversion with a tilted spherical lens has been proven to be highly tilt-angle dependent, especially with high-order topological charge (TC) ⁴⁴.

$$\begin{aligned} \Phi _\alpha = \Phi _\alpha (x,y,k,f_1) = \exp [-\frac{ik}{2f_1}\left( x^2 + \frac{y^2}{\cos ^2(\alpha )}\right) ] \end{aligned}$$

(2)

The tilted lens phase operator $\Phi _\alpha (x,y,k,f)$ depends also on f the focal length, and k the wave-number. For a lens tilted with respect to the x-axis (y-axis), the focus created by the phase along the $y-$axis ($x-$axis) is tighter than the $x-$axis ($y-$axis). The focus along the latter axis remains unchanged. The tilted lens phase can be expressed by Eq. (2). The electric field $E_\alpha (\textbf{r},z=0)$ of the beam right after the lens is written as the output of the operator $\Phi _\alpha$ with $u_{\ell ,p}^{\text {LG}}(r,\varphi ,z=0)$ as the input in Eq. (3).

$$\begin{aligned} E_{\alpha }(\textbf{r},0) = \Phi _\alpha ~u_{\ell ,p}^{\text {LG}}(r,\varphi ,0) \end{aligned}$$

(3)

To obtain the electric field of the transformed LG$_{\ell ,p}$ on the plane (ii), we apply the propagator $\mathcal {P}(\Delta z)$ to $E_\alpha (\textbf{r},0)$ accounting for a propagation distance equal to the focal length of the lens in front of the $\chi ^{(2)}$ crystal ($\Delta z=f_1$).

$$\begin{aligned} E_{\alpha }(\textbf{r},f_1) = \mathcal {P}(\Delta z=f_1)~E_{\alpha }(\textbf{r},0) \end{aligned}$$

(4)

Numerically, the propagation of the beam towards the focal plane (ii) is modeled by calculating the Fresnel integration with the angular spectrum method^46,47. On the plane (ii), the radial symmetry of the input beam is broken with the radial function replaced by a pair of Hermite-Gaussian functions that carry the linear dark stripes of an HG mode corresponding to dislocations with $\pi$ phase jumps.

The third plane−(ii) and (iii) are linked by the up-conversion process in which the incident electric field in Eq. (4) is frequency doubled,

$$\begin{aligned} E_{SH,\alpha }(x,y,z) = E^2_{\alpha }(x,y,z). \end{aligned}$$

(5)

Eq. (5) is a good approximation when using a thin $\chi ^{(2)}$ material and a paraxial-approximated beam. The phase matching is assumed to be perfect and with a non-depleted pump. We omit the effective nonlinear efficiency since we are primarily interested in the linear propagation of the SH beam after the crystal.

The fourth plane−To go from plane (iii) to plane (iv), the electric field in Eq. (5) is propagated to the lens with the propagator operator $\mathcal {P}(\Delta z = f_2)$. At the lens plane, a phase $\Phi _{\alpha =0}(x,y,k,f_2)$ is applied to that beam to mimic the experimental Fourier transform. Then, a second propagator $\mathcal {P}(f_2)$ acting on the last electric field brings it to the Fourier plane, giving the resultant field $E^F_{SH,\alpha }$.

$$\begin{aligned} E^F_{SH,\alpha }(x,y,z+2f_2) = \mathcal {P}(f_2)~\Phi _{\alpha =0}(x,y,k,f_2)~\mathcal {P}(f_2)~E_{SH,\alpha }(x,y,z) \end{aligned}$$

(6)

Eq. (6) illustrates the expression of the electric field on the plane (iv).

Results and discussion

Numerical simulations

We numerically investigate the SHG process of astigmatically transformed LG$_{3,0}$ and LG$_{3,1}$ beams. For both beams, we perform simulations for all planes of interest, from (i) to (iv), at various angle $\alpha$ following our numerical model.

Figure 2a shows the simulated intensity and phase profiles for LG$_{3,0}$ at $\alpha = 0^\circ , 10^\circ$, and $15^\circ$. The intensity and phase profiles of the fundamental beam is imaged Fig. 2a(i). The astigmatically transformed beam is obtained with Eq. (4) and captured at the Fourier plane of the tilted lens as shown in Fig. 2a(ii). The generated second harmonic (SH) signal in the $\chi ^{(2)}$ crystal is taken in the near-field and presented in Fig. 2a(iii). In far-field, the generated SH phase and intensity are calculated according to Eq. (6) and shown in Fig. 2a(iv). The effect of astigmatic transformation is demonstrated in Fig. 2a(ii). At $\alpha = 10^\circ$ and $15^\circ$, astigmatism causing separation of topological charge, generating fringe-like intensity profiles, with dark fringes corresponding to singularities. We refer these profiles as gHLG$^{\alpha }_{\ell ,p}$ mode. While the intensity profile of gHLG$^{15}_{3,0}$ resembles that of a diagonal HG$_{3,0}$, they are distinct due to the lack of topological charge in HG$_{3,0}$ (also see Fig. S1 (ii)). This difference can be interpreted as the introduction of perturbations on the transverse plane by the astigmatic effect that causes the breaking of $\pi$ phase dislocations in HG modes and leads to the formation of vortex streets. More specifically, the difference between two modes is attributed to the tilt lens’s deviation from an ideal $\pi /2$ mode converter. For comparison purposes, the profiles of both beams on all planes of interest are shown in Sect. A1 in Supplementary Materials (SM).

Plane (ii) is connected to plane (iii) through an up-conversion process, which doubles the topological charge. As seen in Fig. 2a(iii), at high $\alpha$, the doubling of OAM creates a triplet of singularities of $\ell =$ 2. Beyond the phase profile, this effect could also be seen in the intensity profile, which shows the widening of singularities, indicating an increase of topological charge. Similar to those on the plane (ii), the singularity positions correspond to vortex centers.

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.

A better visualization can be found in SM Fig. S2. From plane (iii) to plane (iv), at $\alpha = 10^\circ$, and $15^\circ$, further separation occurs, creating phase profiles consisting of 6 vortices center of $\ell =$ 1. This is reflected in the intensity profile as 6 isolated singularities along with a bright line evenly dividing them into a pair of 3 singularities. The thickness of this line increases with a larger value of $\alpha$. To better quantify the angular momentum within the system, we perform calculation of OAM using method introduced by Martinez-Castellanos and Gutiérrez-Vega^48,49. Our input beam, which has the average OAM value $J_z$ = 3.0, is transformed by a titled lens at plane (ii), and underwent SHG process at plane(iii). For numerical simulation, at plane (ii), for all values of $\alpha , J_z$ = 3.0, suggesting that the value of $\alpha$ does not affect the OAM of the beam profile. At plane (iii), for $\alpha = 0^\circ$, 5$^\circ$, 10$^\circ$, 15$^\circ$, $J_z$ = 5.9, 6.0, 5.9, 5.5 respectively. Although these values are approximately close to the expected value of 6.0, the drop of $J_z$ at $\alpha = 15^\circ$ suggests that neither the astigmatic transformation nor the SHG process only create changes in the OAM conservation. However, their combination has a significant effect on the angle momentum conservation, specially at larger tilted angle. We remarks, that $J_z$ values at planes (iv) and (iii) are the same.

We present in Fig. 2b and Fig. S3 the electric field distribution and amplitude profile of the SH signal on the plane (iv) as a function of the increasing tilting angle $\alpha = [0, 15^\circ ]$. Data for electric field distribution are created by combining amplitude and phase profile of the beam. We provide data with overlay of electric field amplitude and phase for better visualizations of positions of vortex centers and singularities. Each vortex center is marked by a white dashed-line circle. At $\alpha = 7^\circ$, vortex centers split evenly into a pair of three. The separation between the pair also increases as a function of $\alpha$, which is reflected in the intensity profiles of the SH signals, shown in Sect. A3 in SM. From the electric field data in Fig. 2b, we observe much clearer separation of these 6 singularities, which are not easily visible in the intensity and phase profiles.

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.

To further characterize the separation and conservation of the topological charge, we numerically investigate the frequency-doubling of LG beams with radial functions for $\alpha \in [0, 15^\circ ]$. Figure 3 details the intensity distribution, phase profile, and electric field data of the generated SH signals in plane (iv) for an input LG$_{3,1}$ beam in plane (i). Detailed results of the pump and the SH signal at planes (i), (ii), and (iii) can be found in SM, A4 and A5.

For $\alpha \le 5^\circ$, Fig. 3 shows that the astigmatic transformation majorly affects the azimuthal function and minimally the radial function. Thus, the azimuthal function is more sensitive to small tilt angles of the spherical lens. At this value, the radial ring of the SH signal’s intensity profile is well maintained, while the azimuthal function is quite similar to the profile shown in Fig. 2a(iv). As the tilt angle continues to increase, the beam profiles continue to evolve to reflect the effect of astigmatic transformation in the radial function (for more details, see SM Fig. S5).

From Fig. 3, we can see in the intensity profile the influence of the radial function on the shape of the inner ring of the SH signal. At sufficiently high $\alpha$, we observe the creation of further singularities beyond the 6 singularities expected for the frequency-doubling of a beam with $\ell$ = 3. At $\alpha = 9^\circ$, Fig. 3, we are able to observe the emergence of these additional singularities at the center of the phase profile, which are marked by a dashed box. From $\alpha = 11^\circ$, these singularities are fully formed, each accompanied by a pair of vortex-antivortex, which maintains the conservation of momentum. As $\alpha$ increases, these pairs of vortex-antivortex not only become more prominent, but also shift from center to outer edge, as depicted in Fig. 3. This dynamic is analyzed at smaller increment of $\alpha$ in SM Sect. A5. Here, we mark the vortex with a white-dash circle, and antivortex with a white cross. We also provide a zoom window of 1 pair vortex-antivortex for electric field data at $\alpha \ge 11^\circ$.

The exact tilted angle ($\alpha _{rm}$) corresponding to the astigmatic transformation of the radial function can be measured by monitoring the appearance of antivortices in the electric field (or in phase). In the intensity profile, the antivortices are more diffuse for small angles, but can be clearly identified for $\alpha \ge 11^\circ$ by the additional singularities created on either side of the beam’s vertical axis of symmetry (see Fig. 3, $\alpha = 13^\circ$). Overall, the numerical model suggest that due to the astigmatic transformation of the radial function, SHG signal possess 2 additional pairs of vortex-antivortex, or 4 additional singularities. Calculation of $J_z$ at plane (ii), (iii), and (iv) is also performed. At plane (ii), $J_z$ = 3.0 for all values of $\alpha$. At plane (iii), for $\alpha = [0^\circ$, 15$^\circ$], the calculated $J_z$ = 5.9, 5.8, 5.5, 5.3, 5.7, 6.0, 6.1. We observe a gradual drop of $J_z$, reaching a minimum at $\alpha = 9^\circ$, before slowly increasing back to the expected value of 6.0. This drop between $\alpha \in [7^\circ , 11^\circ ]$ corresponds to the beginning of the radial mode astigmatic transformation. Our method still shows a helical phase of $\ell = 6$, indicating that the drop in the calculated momentum does not affect the structure of the electric field. In additions, the creation of the pairs of vortex-antivortex does not correspond to the change in value $J_z$. Also the calculated values of $J_z$ at plane (iv) has agreement with values at plane (iii).

Experimental data

We experimentally generate the SH signal following the same process used for the numerically simulated LG$_{3,0}$ and LG$_{3,1}$ beams presented in Fig. 1.

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 4 shows the experimental intensity distribution and phase profile of the LG$_{3,0}$ beam measured using the experimental setup shown in Fig. 7. We observe in Fig. 4a, as expected for $\alpha = 0^\circ$, the frequency doubling of LG$_{3,0}$ on the plane (i) into LG$_{6,0}$ on the plane (iv) with a momentum conservation ($2\ell _\omega \hbar = \ell _{2\omega }\hbar$) confirming our model. As $\alpha$ increases, the change in intensity distribution of the generated beam in plane (iv) shows how the non-commutativity of the operators transformation astigmatic and the SH generation creates new structured beams.

Figure 5

Analysis of singularity’s diameter between gHLG$^{15}_{\ell ,p}$ and its SHG signal. (a) Comparison in line profiles between gHLG$^{15}_{3,0}$ and its SHG signal in numerical model. (b) Comparison in line profiles between gHLG$^{15}_{3,0}$ 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).

The main difference between the numerical simulation in Fig. 2a and Fig. 4a is the orientation of the astigmatically transformed intensity in plane (ii) and its immediate upconversion in plane (iii). This 90$^\circ$ misorientation is due to the axis (x or y) around which the TL is rotated³⁸. In Fig. 4b, we present the reconstructed electric field for $\alpha = 0^\circ$, 5$^\circ$, 10$^\circ$, 15$^\circ$.

Besides, we investigate the relation between signal in plane (ii) and (iii). According to the numerical model, through an SHG process, doubling of topological charges would happen at each singularity independently, increasing it from $\ell = 1$ at plane (ii) to $\ell = 2$ at plane (iii). This is confirmed by the phase profiles shown at Fig. 2.

However, this is not confirmed experimentally due to the lack of phase data at Fourier planes. As such, we investigate this phenomenon using the size of the nil intensity area between non-zero intensity signals at plane (ii) and (iii) for $\alpha = 15^\circ$.

Figure 5a,b plot line profiles following the length of the intensity signal (highlighted by white-dashed lines) at plane (ii) and (iii), both numerically and experimentally, highlighting a larger singularity in plane (iii). Here, singularities are defined by position whose intensity count is 0.05 or less after normalization. Figure 5c delineates the ratio between the singularity diameter at these two planes as a function of topological charge $\ell$. Overall, we could see that the ratios remain $\sim$2 as $\ell$ varies, suggesting the doubling of topological charge upon frequency-doubling. We repeat the calculation of average OAM based on experimentally recorded data at plane (iv). At $\alpha = 0^\circ$, 5$^\circ$, 10$^\circ$, 15$^\circ$, the calculated $J_z$ = 6.1, 5.9, 6.0, 5.3. These values show high agreement with our numerical models.

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).

We also perform experiments to study the behavior of radial functions under the combination of astigmatism transform and frequency doubling process. Figure 6 shows the experimental obtained data for both plane (iii) and plane (iv) with LG$_{3,1}$ as fundamental. As with LG$_{3,0}$, only intensity signals are recorded at plane (iii). For plane (iv), we show intensity profiles, phase profiles obtained from the wavefront sensor, and reconstructed electric field distribution. Experimental SH signals at plane (iii) in Fig. 6 confirm the validity of our numerical model.

Here, we could clearly see the evolution of SHG of gHLG$^{\alpha }_{3,1}$ as a function of $\alpha$, starting with the inner ring to radial function as $\alpha$ increases. In plane (iv), we again track the separation of the vortex centers, which is highlighted by dashed-line circles, due to increased tilting angle $\alpha$. Comparisons between numerical simulation and experimental data reveal slight differences in the locations of topological charges, especially at high $\alpha$. This is due to the reconstruction of the electric field by a wavefront sensor, which only provides phase information in positions where the intensity signal is sufficiently high. As a consequence, data at singularity positions is not available, leading to errors in determining the locations of topological charges.

Nevertheless, we observe a clear trend of increasing separation of singularities at higher $\alpha$ values. We are also interested in identifying the additional pairs of vortex-antivortex due to the astigmatic transformation of the radial function. Compared to Figs. 3, 6 does not show the existence of additional vortex centers in its phase profile at $\alpha = 11^\circ$. However, two pairs of vortex-antivortex are identified for $\alpha = 13^\circ$ and $15^\circ$, with the antivortex center marked by a white cross. In the electric field row, we provide a zoom window in the area of a vortex-antivortex pair for visualization purposes.

To further prove the contribution of the radial functions, we perform simulation and experiment of SHG signals using LG beams with different radial functions, which is discussed in SM Sect. A8. The targeted beams are: LG$_{2,0}$, LG$_{2,1}$, and LG$_{2,2}$. From the phase profiles of these beam, Fig. S8 clearly demonstrate that the number of vortex-antivortex singularities corresponds to the value of radial function of fundamental LG beams.

We describe this process as a mitosis-like division due to its resemblance to the metaphase, anaphase, and telophase stages of cell division. The doubling of the topological charge, or the conservation of orbital angular momentum (OAM), produces a SH beam with a spatial distribution analogous to DNA alignment in metaphase, followed by a splitting behavior akin to anaphase. These dynamics are illustrated in Figs. 2 and 4: From plane (i) to (ii), astigmatic transformation aligns singularities along a diagonal line, mirroring metaphase. From plane (ii) to (iii), the topological charges at these singularities double, analogous to DNA replication in interphase. From plane (iii) to (iv), as the SH signal propagates, the singularities split symmetrically, each retaining a topological charge of $\ell$, resembling the separation of chromosomes in anaphase and telophase. Lu et al.⁵⁰ previously discussed a phenomenon in which, through an SHG process, the split of output profile behaves like binary fission process in a prokaryotic cell. This work mainly concerned with the dynamic of polarization in the frequency-doubling process. In comparison, the main focus of the phenomenon described in our work is the orbital angular momentum in an SHG process. It is the division of the singularity (topological charge) that drives the formation of novel beam profiles when noncommutative operators are combined. The discovery of this beam profile could have great impact for studies looking to perform transfer of OAM with stronger emphasis on localized effect. Recent works^51,52 have demonstrated the ability to apply twisted strain onto ferroelectric material using LG beam to create and destroy topological polar structures. Instead of a conventional LG beam, a similar study that utilizes the electric field presented here would be able apply this twisted strain at multiple position within the ferroelectric sample simultaneously. Additionally, Kim et al.⁵³ also discussed the assemble of chiral nanostructure under the influence of circularly-polarized light. A potential follow-up would be to investigate the chirality of nanostructure whose growth driven by this novel beam profile.

Conclusion

In Summary, we have experimentally demonstrated, supported by a numerical model and phase retrieval algorithm, the generation of unconventional structured light via astigmatic transformation and up-conversion of LG beams. While it is known that LG modes can be converted into diagonal HG modes using a tilted spherical lens, we leverage the angular dependence of the output mode’s intensity distribution to produce novel SH profiles in both intensity and phase. Strikingly, the evolution of the SH light exhibits strong parallels with certain stages of cell mitosis. We propose that such structured beams could be used to probe mitosis dynamics, particularly in studying chromosome movement, where the underlying mechanical forces remain insufficiently understood.

Further, recent work⁵⁴ has shown that up-converting circularly polarized optical vortices can yield unexpected topological charges due to the conservation of both spin and orbital angular momentum. Extending our approach to circularly polarized LG beams under astigmatic transformation could provide deeper insights into angular momentum dynamics.

Beyond biophysics, we anticipate broad applications for these structured beams due to their unique phase and intensity profiles. For instance, optical vortices have been employed to control domain evolution in ferroelectrics⁵² or manipulate biomolecules as optical tweezers⁵⁵. Unlike conventional LG beams, our SHG signals feature isolated singularities, enabling localized OAM transfer, offering finer control compared to the uniform effect of standard Laguerre-Gaussian modes.

Method

Experimental setup

To generate experimentally the beams illustrated in Fig. 1, we implement an optical setup whose schematic is shown in Fig. 7. An expanded and collimated femtosecond mode-locked regenerative Ti:Sapphire laser centered at 808 nm with a repetition rate of 3 kHz and a pulse energy and duration of 1.67 mJ and 38 fs respectively, is highly attenuated by a neutral density filter before being reflected by an SLM.The SLM is a Santec SLM-200 with resolution $1080 \times 1920$ pixels, and pixel size of 8 $\upmu$m. The beam is expanded by a pair of concave and convex lenses to avoid the formation of a filament.

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.

An iris is introduced to create a nearly flat wavefront. The polarization of the light is modulated by a half-wave plate such that it matches the preferred polarization of the SLM. Different structured light profiles are generated by the SLM using phase masks with diffraction grating, which provides complex amplitude modulation of the field ^7,8.

As shown in Fig. 7, a second iris is used to select the first order at the far field. The LG$_{\ell , p}$ beam is imaged in plane (i). To image the beam profile at plane (ii), a beam splitter (BS), not included in Fig. 7, is placed after the tilted lens (f$_{TL}$ = 10 cm). This system allows us to image dynamically the astigmatic transform of the LG beam using a tilted spherical lens mounted on a metric rotational stage. The newborn gHLG beam is incident on a 0.5 mm thick type-I $\beta$–Barium Borate (BBO) crystal to be upconverted. The generated SH signal is captured by a lens (L$_1$), of focal length f$_{1}$ = 15 cm, before being filtered by a $405\pm 10$ nm bandpass filter. The reason for choosing f$_{1}$ > f$_{TL}$ is to increase the resolution of signal at plane (iv) for higher accuracy from the detectors. Detector 1, a scientific Complementary Metal–Oxide–Semiconductor (sCMOS) camera, is used to image in far-field the intensity profile of the SH signal in plane (iv). The sCMOS camera has resolution 1080x1920 pixel, with pixel size of 5.04$\mu$m. The sCMOS camera is replaced by a high spatial sampling wavefront sensor (HASO4 126 Broadband) to collect the phase information of the beam at planes (i) and (iv). Our wavefront sensor has resoluion 170x126, with pixel size of $\sim$80 $\mu$m.

SH signals at plane (iii) and (iv) are connected by a spatial Fourier transform. The relation is achieved through a spherical lens (L$_1$). To image the intensity profile of the SH beam immediately after the crystal, a second lens (L$_2$), whose focal length is f$_{2}$ = 15 cm, is placed to create, together with L$_1$, a 4f imaging system. Another sCMOS camera is placed at the L$_2$ focal length to image the intensity profile at plane (iii), as shown in Fig. 7. Only the intensity profiles were experimentally recorded for planes (ii) and (iii) since the beam size is too small to be spatially sampled by the wavefront sensor.

Generation of electric field data

Visualization of electric field distribution shown in Figs. 2, 3, 4, 6 is generated using Python. For numerical model, amplitude and phase data can be extracted directly from numerical results at plane (iv). For experimental results, phase distribution is reconstructed from the wavefront data collected by the Wavefront Sensor. Amplitude data is reconstructed by taking the square-root of the intensity data collected by the Wavefront Sensor.