Tunable-Charge Optical Vortices Through Edge Diffraction of a High-Order Hermit-Gaussian Mode Laser

Abstract

An optical vortex is a typical structured light field characterized by a helical wavefront and a central phase singularity. With its expanding applications in modern information technology, the demand for generating vortex beams with diverse topological charges continues to grow. Existing methods for modulating the topological charges of vortex beams involve complex operations and high costs. This study proposes a novel approach to modulate the topological charges of optical vortices through edge diffraction of a high-order Hermit–Gaussian (HG) mode laser. First, a high-order HG mode laser is built using off-axis pumping configuration. By selectively obscuring specific lobes of the high-order HG beam, various optical vortices are generated using a cylindrical lens mode converter. The topological charge can be continuously tuned by controlling the number of obscured lobes. This method substantially improves the efficiency of topological charge modulation, while also enabling the generation of fractional vortex states. These advancements show potential in mode-division-multiplexed optical communications and encryption.

1. Introduction

An optical vortex with a quantized topological charge is a typical structured light beam that carries orbital angular momentum (OAM) [1], characterized by a helical wavefront [2] and a central phase singularity. Owing to this unique property, vortex beams [3] have found expanding applications in optical communications [4,5], optical tweezers [6,7,8,9], imaging [10,11,12], sensing [13,14], and other fields [15], where the manipulation of topological charges [16] has played critical roles. For instance, in optical communications, employing vortex beams with various topological charges as independent channels for parallel data transmission can enhance the capacity by tens to hundreds of times [5]. In holography, vortex beams with distinct topological charges provide independent information channels, enabling multidimensional and high-security information encryption [17,18]. Therefore, the generation of diverse high-order vortex beams holds significant potential in modern information technology.

Currently, the topological charges of optical vortices are mainly manipulated [19] using extra-cavity or intra-cavity methods. Extra-cavity-based methods predominantly employ spatial phase elements, such as a spatial light modulator (SLM) [20] or a spiral phase plate (SPP) [2]. SLMs are characterized by high manufacturing costs and relatively slow response speeds due to the intrinsic response time of liquid crystal molecules. SPPs generate optical vortices by controlling the optical path difference through varying material thickness. This fundamental principle dictates that a single SPP can only generate a single vortex beam with a fixed topological charge [21,22]. Alerting the topological charge requires physical replacement of the element. By tailoring the cavity structure, laser transverse modes with controllable orders can also be directly generated within the resonator [23]. Typical methods include off-axis pumping [24] and etching patterns on cavity mirrors [25]. Off-axis pumping technology enables continuous modulation of the topological charges of vortex beams. Nevertheless, this approach requires precise control of the pump size and position, which limits its flexibility. As for the mirror surface etching technique, the topological charge is adjusted by physically moving the cavity mirror to align with a different etching spot. This operation is difficult to achieve continuous modulation of the topological charges.

In this work, we propose a simple method for tuning the topological charges of optical vortices via edge diffraction using a fixed high-order Hermit–Gaussian (HG) mode laser. Firstly, a high-order HG beam is generated directly from a laser through the off-axis pumping method. Subsequently, by selectively obscuring specific lobes of the high-order HG beam using an aperture, optical vortices with various topological charges are produced using a single-cylindrical-lens mode converter. The topological charge can be continuously tuned by adjusting the number of obscured lobes. This work employs a fixed laser output mode to generate optical vortices with tunable orders in a simple and efficient manner, significantly improving the flexibility and efficiency of topological charge modulation.

2. Materials and Methods

2.1. Experimental Setup

To generate optical vortices with continuously tunable topological charges, an experimental setup was constructed, as illustrated in Figure 1. A high-order HG mode laser was built using an off-axis pumping scheme. In detail, an 808 nm pump beam was collimated by lens L₁ and focused through lens L₂ into an Nd: YVO₄ crystal. The focal lengths of L₁ and L₂ were 100 mm and 125 mm, respectively. As shown in Figure 1a, the resonator consists of a concave mirror M₁ with a curvature radius of −100 mm, an output coupler (OC) with a transmittance of 2%, and a Nd: YVO₄ crystal with 3 × 3 × 5 mm³ and a doping concentration of 0.5%. By tilting M₁ to achieve off-axis pumping, a high-order HG beam was generated. Firstly, the HG mode was converted to an optical vortex using a single-cylindrical-lens mode converter. Thereafter, a single-side aperture was introduced to obscure a specific number of lobes of the HG beam to facilitate a lower-order transverse mode. Subsequently, a lower-charge optical vortex was obtained. In order to characterize the spatial phase of an optical vortex, a Mach–Zehnder interferometer was implemented by splitting the output laser beam into two paths [26]. In the upper path, an optical vortex was obtained using a single-cylindrical-lens mode converter. In the bottom path, one lobe of the high-order HG mode beam was used to interfere with the optical vortex. In the two paths, several lenses were used to control the beam sizes, in which the focal lengths of L₃, L₄, L₅, L₆, L₇, and L₈ were 300 mm, 500 mm, 50 mm, 300 mm, 500 mm, and 50 mm, respectively. In addition, the focal length of the cylindrical lens (CL) was 50 mm. The intensity distributions of HG modes, vortex modes, and interference patterns were recorded using a CCD camera.

2.2. Simulations

In this work, optical vortices with various topological charges were obtained via edge diffraction using a fixed high-order HG mode beam, where an aperture was used to adjust the topological charges and a cylindrical lens was used to generate optical vortices. We employed the diffraction integral method to simulate the processes. In detail, the field distribution of a diffracted obscured HG mode can be expressed as follows:

$$U_{\mathrm{in}}(x,y;z)={\displaystyle \frac{e^{ikz}}{i\lambda z}}\iint U_0(x^{\prime },y^{\prime })\exp \left({\displaystyle \frac{ik}{2z}}\left({(x-x^{\prime })}^2+{(y-y^{\prime })}^2\right)\right)d{x}^{\prime }d{y}^{\prime },$$

(1)

where the obscured HG mode can be described as the following formula:

$$\begin{array}{c}{U}_0(x^{\prime },y^{\prime })=E_0{\displaystyle \frac{w_0}{w(z_0)}}{H}_m\left({\displaystyle \frac{\sqrt{2}{x}^{\prime }}{w(z_0)}}\right)H_n\left({\displaystyle \frac{\sqrt{2}{y}^{\prime }}{w(z_0)}}\right)\exp \left[-{\displaystyle \frac{x^{{\prime }^2}+y^{{\prime }^2}}{w^2(z_0)}}-i{\displaystyle \frac{k(x^{{\prime }^2}+y^{{\prime }^2})}{2R(z_0)}}\right]\\ \times \exp \left[\begin{array}{c}i{\psi }_{m,n}(z_0)\end{array}\right]e^{-ik{z}_0}A(x^{\prime }-x_c)\end{array},$$

(2)

In Equation (1), $U_{in}(x,y;z)$ is the complex amplitude at the front surface of the cylindrical lens; $z$ is the propagation distance; $U_0(x^{\prime },y^{\prime })$ is the complex amplitude after aperture modulation; $k$ is the wavenumber; and λ is the wavelength. In Equation (2), $z_0$ is the axial distance from the beam waist to the aperture; $E_0$ is the overall amplitude coefficient; $w_0$ is the waist radius; $w\left(z_0\right)=w_0\sqrt{1+{(z_0/z_R)}^2}$ is the beam radius at $z_0; H_m\left[{\displaystyle \frac{\sqrt{2}{x}^{\prime }}{w\left(z_0\right)}}\right] \mathrm{a}\mathrm{n}\mathrm{d} H_n\left[{\displaystyle \frac{\sqrt{2}{y}^{\prime }}{w\left(z_0\right)}}\right]$ are the Hermite polynomials of orders m and n, respectively; $z_R=\pi w_0^2/\lambda$ is the Rayleigh length, $R\left(z\right)=z[1+{(z_R/z)}^2]$ is the radius of curvature of the equal-phase surface, and the Gouy phase is ${\psi }_{m,n}\left(z\right)=\left(m+n+1\right) \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n} (z/z_R)$. Specifically, $A(x^{\prime }-x_c)$ is the Heaviside step function for a single-sided aperture, which equals 1 when $x^{\prime }>x_c$ and equals 0 when $x^{\prime }<x_c$.

A cylindrical lens was used to convert the obscured HG mode to an optical vortex, in which the power axis of the cylindrical lens is aligned at a 45° angle with respect to the x-axis. After conversion, the field distribution can be expressed as the following equation:

$$\begin{array}{l}{U}_f(X,Y)={\displaystyle \frac{e^{ikf}\cos \theta }{i\lambda f}}\exp \left[{\displaystyle \frac{ik}{2f}}{(X\cos \theta +Y\sin \theta )}^2\right]\\ \times \int {\int }_{-\infty }^{+\infty }{U}_{in}(x,y)\exp \left[-i{\displaystyle \frac{2\pi }{\lambda f}}(x\cos \theta +y\sin \theta )(X\cos \theta +Y\sin \theta )\right]dxdy\end{array}.$$

(3)

In Equation (3), $U_f(X,Y)$ is the complex amplitude at the focal plane of the cylindrical lens, $\theta =45°$, and $f$ is the cylindrical lens focal length. Under the above conditions, an obscured HG mode is converted into an optical vortex through astigmatic transformation. In the process of mode transformation, the HG mode laser passes through a cylindrical lens and records in the far-field plane. The diffraction process satisfies the Fourier transform from spatial frequency domain to spatial domain. Thus, we utilized the Fourier transform method to simulate the diffraction.

We calculated the expectation value of OAM to quantify the topological charge. The expectation value of OAM can be expressed as follows:

$$\langle L_z\rangle ={\displaystyle \frac{\iint U_f^{*}({\widehat{L}}_z{U}_f)dxdy}{\iint |U_f{|}^{2}dxdy}}.$$

(4)

In Equation (4), $⟨L_z⟩$ is the expectation value of OAM in the z direction; ${\widehat{L}}_z(=-i\mathrm{\hslash }\partial /\partial \mathrm{\varnothing })$ is the OAM operator, where $\mathrm{\hslash }$ is the reduced Planck constant; and $\mathrm{\varnothing }$ is the azimuth angle. Consequently, the topological charge l can be obtained as follows:

$$l={\displaystyle \frac{Im\left(\iint U_f^{*}(\frac{\partial U_f}{\partial \varphi })dxdy\right)}{\iint |U_f{|}^{2}dxdy}},$$

(5)

where $Im\left(\right)$ is the operation for extracting the imaginary part.

3. Results

First, we built a high-order HG mode laser and demonstrated the continuous modulation of the topological charges of optical vortices from l = 2 to l = 1. By tilting the concave mirror M₁ to a specific angle, a HG_0,2 mode beam was generated. Using a cylindrical lens, an optical vortex with the topological charge l = 2 was obtained. The intensity distribution of the optical vortex of l = 2 can be observed to exhibit a uniform ring shape. By selectively obscuring the top lobe of the HG_0,2 mode beam, the beam was modulated into the TEM_0,1 mode. With the CL removed, the intensity distribution of the TEM_0,1 was recorded after the beam was focused by L₅, as shown in Figure 2a. The corresponding converted optical vortex with the topological charge l = 1 is presented in Figure 2b. The intensity distribution of this optical vortex is non-uniform, with weak intensity on the right side. In contrast to the optical vortex with l = 2, the dark central spot of this beam is substantially smaller. To characterize the spatial phases, the built Mach–Zehnder interferometer was used to produce the interference between an optical vortex and a near-plane wave. The interference patterns are shown in Figure 2c. For the optical vortex with l = 2, the interference pattern presents two bifurcations in the center. For the optical vortex with l = 1, the interference pattern exhibits one bifurcation in the center. As such, the interference results demonstrated the tuning of the topological charge from l = 2 to l = 1. We also retrieve the corresponding phase distributions according to the interference patterns, which further clearly reveal the helical phases with l = 2 and l = 1, respectively.

Numerical simulations were conducted to further reveal the mechanism for tuning the topological charges, as shown in Figure 2e–h. For the unobstructed HG_0,2 mode, the phase difference between adjacent lobes is π. This means that the phase transits twice by an amount of π across the transverse plane. As such, after conversion using a CL, the phase distribution of the converted optical vortex is characterized by two full 0 to 2π progressions per single azimuthal rotation. In contrast, when the top lobe of the HG_0,2 mode is rightly obscured, the obtained TEM₀,₁ mode undergoes only single 0 to π phase shift along the transverse direction. This results in the phase distribution with only one cycle per revolution. Moreover, the simulated intensity distribution of the optical vortex with l = 2 exhibits a uniform ring structure, whereas that of the optical vortex with l = 1 is non-uniform. Both the intensity distributions and phase distributions are in good agreement with the experimental results. Therefore, these results demonstrate the feasibility of the proposed edge diffraction method for tuning topological charges.

Next, we demonstrated the modulation across an even broader range of topological charges using a higher-order HG beam laser. By tilting the concave mirror M₁ to a larger angle, we generated a HG_0,5 mode beam. Using the aforementioned edge diffraction method, a continuous transition from HG_0,5 to TEM_0,4, TEM_0,3, TEM_0,2, and TEM_0,1 was achieved and recorded by a CCD after propagating a certain distance, in which a TEM_0,n mode has n lobes, as shown in Figure 3a. The transverse modes were then converted to optical vortices from l = 5, l = 4, l = 3, l = 2, and l = 1, respectively, as shown in Figure 3b. The intensity distribution of the optical vortex with l = 5 appears as a uniform ring, while those of the modulated optical vortices with l = 4, l = 3, l = 2, and l = 1 show non-uniform distributions. The ring diameters of optical vortices gradually decrease with the reduction in their topological charges. The observed effect is attributed to the lower orbital angular momentum (OAM) carried by a lower-order optical vortex. The diminished OAM manifests as a gentler wavefront phase variation, thereby confining the energy closer to the optical axis and resulting in a smaller ring diameter. The interference patterns of the optical vortices are shown in Figure 3c. The numbers of fringe bifurcations correspond to the topological charges of the optical vortex, which indicate that we have successfully generated optical vortices with l = 4, l = 3, l = 2, and l = 1. Thereby, the results confirmed the feasibility of the proposed method to achieve continuous modulation of topological charges over an extended range. Note that the modulated TEM₀,ₙ beams tend to recover to regular HG_0,n mode with propagation, as shown in Figure 3a. This evolution can be attributed to the self-healing property of the HG mode after edge diffraction [27].

Furthermore, we also compared the optical vortices generated by obstructing a HG beam from different sides. A HG_0,5 mode beam was employed, and optical vortices with l = 3 were created by blocking two lobes from the top, bottom, and both sides simultaneously (Figure 4). The results confirm that the topological charge is determined solely by the number of obscured lobes. However, the intensity distribution of the vortex is highly sensitive to the obscuration ways: top and bottom obstructions lead to weak-intensity regions on the right and left, respectively, while dual-side obscuration reduces intensity on both sides. This demonstrates the possibility of independently controlling the topological charge and the intensity distribution of the optical vortex.

Finally, we recorded the optical field distributions of optical vortices under varying degrees of obscuration applied to a HG beam. By gradually obscuring the top lobe of the HG₀,₅ mode, the evolution of the vortex field was recorded. We observed that when one lobe of a HG beam is partially obstructed, it generates an optical vortex with an unclosed ring, as shown in the middle column of Figure 5b. As the aperture is continuously moved until it completely blocks the entire lobe, the ring evolves from an unclosed to a closed ring profile. We hypothesize this phenomenon of an unclosed ring to be attributed to the generation of a fractional-charge vortex mode [28,29,30]. To verify this conjecture, we collected and analyzed its interference pattern. As shown in the central part of Figure 5c, the interference fringes exhibit distinct dislocations, which satisfy the typical features of fractional vortex beams and confirm our hypothesis. The region of fringe dislocation indicates an area of phase discontinuity, resulting in an unclosed ring. Therefore, the partially blocked state is responsible for both the generation of a fractional vortex and the non-closure of the ring. The right half of Figure 5 presents simulation results, which show good agreement with the experimental observations. The fractional vortex topological charge was calculated to be 4.23.

It is worth noting that, while the distributions of the phase singularities in both the vortices with l = 5 and l = 4 exhibit relative concentration, the phase singularities in the fractional vortex display significant dispersion (Figure 5f). As depicted in Figure 5e, the intensity distribution of the optical vortex with l = 4 is non-uniform yet continuous. However, the intensity distribution of the fractional optical vortex is discontinuous. We simulated the intensity at the notch position in the vortex beam as the function of obstruction position, as depicted in Figure 6. When a part of a lobe in a HG mode is blocked, the obtained optical vortex can be regarded as the coherent superposition between an integer-order vortex beam and the remaining partial lobe. As such, the destructive interference results in the intensity singularity in the ring.

4. Discussion

Through the experiments, we demonstrated a dynamic method to modulate the topological charges of optical vortices by selectively blocking parts of high-order HG beams. Compared to existing methods such as SPP or cavity structure adjustment, this approach demonstrates superior flexibility and cost efficiency. Additionally, digital micromirror devices (DMDs) can operate at frequencies of several kHz when performing such tasks, while their cost is significantly lower than that of spatial light modulators (SLMs). By employing a DMD as the aperture in our system, the topological charge can be switched much faster.

In our system, the topological charge can be continuously adjusted between two adjacent integers while continuously obscuring one lobe of a HG mode. Under the current experimental conditions, the minimum step size for each adjustment is about 10 μm, and the size of one lobe is approximately 170 μm. As such, the average adjustment resolution of the topological charge is about 0.06. This capability of continuous modulation of topological charges can be applied to the coding space [31], which can significantly enhance the channel capacity of optical communication. Fractional-order vortices exhibit this missing phenomenon in their light intensity. It is equivalent to the interference between a vortex beam carrying an integer topological charge and the remaining partial lobes after obstruction. The intensity singularity is the result of destructive interference. Figure 6 shows the situation of missing intensity in fractional-order vortices. On top of that, the experiment in this study was performed based on a 1064 nm Nd: YVO₄ laser. It is worth emphasizing that the proposed method is universal and can be extended to other common laser systems, such as a Ti: Al₂O₃ laser and erbium-doped fiber laser. The mode transformation between a HG and a LG mode can also be achieved using a tilted lens [32] or an elliptically squeezed axicon [33].

The proposed method exhibits the limitation of energy loss, although the method is simple. In order to quantify the energy loss of this method, we calculated the loss rates under various blocking depths of a fixed HG₀₅ mode. The energy loss is 29.44% when one lobe is blocked, 40.73% when two lobes are blocked, 50.00% when three lobes are blocked, 59.27% when four lobes are blocked, and 70.56% when five lobes are blocked. In addition, the generated vortex beams deviate from standard LG modes. This deviation arises because the obstruction of a HG mode disrupts its original spatial symmetry, which consequently breaks the spatial symmetry of the vortex beam generated through the mode conversion. Such vortex modes are difficult to apply in optical fiber communications. Nevertheless, they still hold practical potential in specific application scenarios such as optical encryption and free-space optical communications, where the encoding dimension is the topological charge itself rather than the precise intensity distribution. In addition, vortex beams with asymmetric intensity distributions also have potential applications in optical tweezers [34], where the rotating motion of a microparticle can be precisely manipulated in a local area.

5. Conclusions

This study demonstrates that introducing controllable spatial obscuration into the propagation path of a high-order HG beam enables continuous and dynamic modulation of the topological charges of optical vortices. Experimental results show that localized obscuration in the HG beam induces reconstruction of the diffraction field during propagation, thereby generating an optical vortex with the desired topological charge at the output. In addition, the intensity distribution under a fixed topological charge can also be manipulated by governing the obscuration position of the HG mode. Furthermore, by adjusting the obscuration area ratio of a single lobe, we also demonstrated the generation of a fractional optical vortex. Compared with traditional methods, the edge-diffraction-based modulation mechanism does not require complex optical alignment or phase elements, thereby reducing system complexity, enhancing modulation efficiency, and enabling dynamic control.