High-order Hermite-Gaussian Research Articles

Experimental Verification of the Gouy Phase for Higher-Order Hermite-Gaussian Beams

The Gouy phase is related to the axial phase shift experienced by any focused light beam with respect to a reference plane wave when passing through its focus. Generally, the Gouy phase ΦG for a higher-order Hermite-Gaussian (HG) beam is this for the fundamental Gaussian beam multiplied by a factor, which is the sum of the mode indices of the HG beam plus unity. Although this result is a paradigm in optics, we are not aware of its experimental verification. In this paper, we present experimental results obtained with a single-lens interferometer for higher-order HG modes generated using spatial light modulator. The retrieved Gouy phase ΦG is found in a very good quantitative agreement with the mentioned theoretical result.

Analysis of Factors Influencing the Generation of a Higher-Order Hermite–Gaussian Mode Based on Cascaded Spatial Light Modulators

Analysis of Factors Influencing the Generation of a Higher-Order Hermite–Gaussian Mode Based on Cascaded Spatial Light Modulators

Controlled generation of OAM-switchable vortex beams at 639 nm from a Pr3+: YLF laser

In this paper, we reported and demonstrated a blue LD-pumped Pr3+: YLF laser for the controlled generation of orbital angular momentum (OAM) switchable vortex beams at 639 nm. To begin with, the quantitative tilt imbalance of the resonator was introduced to select the higher-order Hermite-Gaussian (HG) modes by utilizing the eigenvector method and introducing the concept of mode discriminant factor. Moreover, the mode converter, composed of two cylindrical mirrors, was applied to convert the Hermite- Gaussian (HG) modes into the Laguerre-Gaussian (LG) modes with the OAM. Then, the experiment setup was built to validate the correctness and effectiveness of the quantitative tilt imbalance. The results showed that the vortex beams with OAM of 0 to ±14 at 639 nm can be obtained by adjusting the tilted angle of the output mirror, which agreed well with the simulation results, and the slope efficiencies of the vortex beams with OAM = 1, 2, and 3 are 11.64%, 8.01%, and 6.05%, respectively. The method provided a viable solution for the generation of visible vortex beams with switchable OAM from the Pr3+: YLF laser.

Eigen-function division multiplexed coherent optical transmission in time domain by using higher-order Hermite-Gaussian pulses.

We describe a new multiplexing technique and its application to demultiplexing in the time domain by using higher-order Hermite-Gaussian (HG) pulses, which are solutions of the Schrödinger equation. We call this technique eigen-function division multiplexing (EDM). This method enables us to further increase the total transmission capacity by superimposing many different HG pulses in the same time slot. This technique is different from a conventional optical time domain multiplexing (OTDM) technique using interleaving, where one pulse exists only in one time slot. The transmitted EDM HG pulses can be demultiplexed by adopting the time-domain orthogonality of the HG pulses (eigen-function orthogonality). The information carried by the mth-order HG pulse (HGm pulse) can be coherently detected by a photo detector, where photo-mixing with a phase-locked HGm pulse generated by a local oscillator can realize demultiplexing. The overlap integral with a different HG pulse becomes zero due to the time domain orthogonality. First, we show numerically that such a new EDM transmission scheme in the time domain is possible. We then show experimentally that we could successfully carry out an EDM HG coherent pulse transmission with four different HG pulses (HG0, HG1, HG2, and HG3), where we report a 400∼480 Gbit/s (10 Gbaud x 4 eigen-functions x 2 pol-mux.) 32∼64 QAM EDM transmission over 300∼450 km.

A Systematic Summary and Comparison of Scalar Diffraction Theories for Structured Light Beams

Structured light beams have recently attracted enormous research interest for their unique properties and potential applications in optical communications, imaging, sensing, etc. Since most of these applications involve the propagation of structured light beams, which is accompanied by the phenomenon of diffraction, it is very necessary to employ diffraction theories to analyze the obstacle effects on structured light beams during propagation. The aim of this work is to provide a systematic summary and comparison of the scalar diffraction theories for structured light beams. We first present the scalar fields of typical structured light beams in the source plane, including the fundamental Gaussian beams, higher-order Hermite–Gaussian beams, Laguerre–Gaussian vortex beams, non-diffracting Bessel beams, and self-accelerating Airy beams. Then, we summarize and compare the main scalar diffraction theories of structured light beams, including the Fresnel diffraction integral, Collins formula, angular spectrum representation, and Rayleigh–Sommerfeld diffraction integral. Finally, based on these theories, we derive in detail the analytical propagation expressions of typical structured light beams under different conditions. In addition, the propagation of typical structured light beams is simulated. We hope this work can be helpful for the efficient study of the propagation of structured light beams.

Tilt Measurement at the Quantum Cramer–Rao Bound Using a Higher-Order Hermite–Gaussian Mode

The quantum Cramer–Rao bound (QCRB) provides an ultimate precision limit in parameter estimation. The sensitivity of spatial measurements can be improved by using the higher-order Hermite–Gaussian mode. However, to date, the QCRB-saturating tilt measurement has not been realized. Here, we experimentally demonstrate tilt measurements using a higher-order HG40 mode as the probe beam. Using the balanced homodyne detection with an optimal local beam, which involves the superposition of high-order HG30 and HG50 modes, we demonstrate the precision of the tilt measurement approaching the QCRB. The signal-to-noise ratio of the tilt measurement is enhanced by 9.2 dB compared with the traditional method using HG00 as the probe beam. This scheme is more practical and robust to losses, which has potential applications in areas such as laser interferometer gravitational-wave observatories and high-sensitivity atomic force microscopes.

Propagation Characteristics of Hermite–Gaussian Beam under Pointing Error in Free Space

Hermite–Gaussian (HG) beams have significant potential to improve the capacity of free-space optical communication (FSOC). The influence of pointing error on the propagation characteristics of an HG beam cannot be ignored in the FSOC system. Although the average irradiance of the HG beam under a small pointing error from the FSOC tracking mechanism has been investigated through Taylor series approximation, here, we propose that the average irradiance of the HG beam under an arbitrary magnitude pointing error can be deduced through a statistical averaging method. We firstly found that the average irradiance profile of an HG beam finally changes into an approximately Gaussian shape with the increase in pointing error and propagation distance and a larger beam waist at the transmitter could mitigate the profile change. The correlation coefficient between deduced theoretical expression and Monte Carlo simulation reaches 0.999. Additionally, the effective spot size, location of the local extreme value, average received power and signal-to-noise ratio (SNR) loss for an HG beam under pointing error were theoretically deduced and analyzed for the first time. We found that the effective spot size of the higher-order HG beam experiences less broadening under the pointing error than that of the lower-order HG beam. The fundamental theoretical expressions of average irradiance for an HG beam under pointing error have provided effective guidance for analyzing the propagation characteristics and link performance.

Generation of Higher-Order Hermite–Gaussian Modes via Cascaded Phase-Only Spatial Light Modulators

The spatial distribution of higher-order Hermite–Gaussian (HG) modes is more complicated than the fundamental mode, and the characteristics of different modes and their orthogonal characteristics have essential applications in the fields of measurement, imaging, and large-capacity communications. The main issue in future applications is how to efficiently generate higher-order HG modes. The spatial light modulator has high spatial resolution and excellent maneuverability, so it is a simple and effective way to generate higher-order HG modes. Here, the efficient generation of higher-order HG modes via cascaded spatial light modulators was experimentally investigated. For the HG90 mode, the conversion efficiency is 61%. The method with high conversion efficiency has promising application potentials in biophotonics, laser physics, and quantum information.

Generation of one-dimensional high-order Hermite-Gaussian laser beams with large mode volume

Hermite-Gaussian (HG) beams have many important applications in the optical frontier, and the limited output power of the high-purity HG beams is partly due to the small gain volume of the mode. The commonly used off-axis end-pumped scheme offers a narrow gain volume whose diameter is about a hundred microns. In this work, a new method of generating the HG beams based on a slab resonator that has a large mode volume is proposed and experimentally demonstrated. According to the optical resonator theory, the intra-cavity modes in thickness and width direction of the slab resonator are restricted by inserting two size-adjustable apertures, respectively. The one-dimensional HG beam generation is mainly guaranteed by the size of the aperture along the thickness direction of the slab, which matches the diameter of the fundamental mode. The different order one-dimensional HG beams are obtained by refined intra-cavity mode modulation. Since the higher-order modes are less sensitive to the misalignment of the cavity mirror than the lower-order modes, and the manipulation of the modes-loss at different orders is achieved by combining the tilt control of the coupled output mirror and the size control of intra-cavity apertures. By adjusting the optical gain and loss in the resonant cavity, the single mode wins the competition of laser modes. Therefore, high-purity one-dimensional HG beams with 0 to 9 orders (HG<sub>00</sub> to HG<sub>09</sub>) are generated. The pump module is comprised of a two-dimensional laser diode array which offers face-pumping to the large surface of the slab, therefore the width of the mode volume is extended to several millimeters. By further incorporating the 100mm-level long slab, the total gain volume is much larger than the counterpart in the off-axis pumping scheme. In this work, the output power of the highest order HG<sub>09</sub> mode increases up to 244 mW. Owing to the large gain volume and uniform gain distribution caused by the face-pumped slab, the purity of high order HG modes is quite good. The correlation coefficient <inline-formula><tex-math id="M3">\begin{document}$ \rho $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20221422_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20221422_M3.png"/></alternatives></inline-formula> between the measured intensity distribution and the theoretical value is larger than 0.95. The beam quality factor <inline-formula><tex-math id="M4">\begin{document}$ {M}^{2} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20221422_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20221422_M4.png"/></alternatives></inline-formula> is also in good agreement with the theoretical one. Finally, a conversion from Hermite-Gaussian beams to the donut-shaped Laguerre-Gaussian beams is realized by using an astigmatic mode converter. Hopefully, power scaling of the HG beam output is also expected by employing cascaded slab amplifiers, and the approach in this paper provides a novel solution for generation of high power HG beams.

Direct generation of ultrafast vortex beam from a Tm:CaYAlO4 oscillator featuring pattern matching of a folded-cavity resonator.

Optical vortices, beams with spiral wavefronts and screw phase dislocations have been explored in applications in optical manipulation, quantum optics, and the next generation of optical communications. In traditional methods, optical vortices are generated using space light modulators or spiral phase plates, which would sharply decrease the integration of optical systems. Different from previous transverse mode conversion outside the cavity, here we experimentally demonstrate a direct generation of ultrafast vortex beam from a Tm:CaYAlO4 oscillator by pattern matching of a six-mirror-folded-cavity resonator. By accurately adjusted the angle of the end mirror and the distance L between the M4 and the SESAMs to control the beam diameter of laser incidence on the gain medium in the sagittal and tangential planes, a stable 2 µm ultrafast vortex laser emission of annular Laguerre-Gaussian (LG) mode was obtained with a maximum output power of 327 mW and pulse duration of 2.1 ps. A simple YAG crystal plate was used as handedness selector and a homemade Mach-Zehnder (MZ) interferometer has verified the vortical property of the LG01 mode. By furtherly controlling the cavity mode pattern matching, other stable transverse-mode operations for TEM00, high-order Hermite-Gaussian (HG) transverse mode and doughnut-shaped beams were successfully realized. This work provides a flexible and reliable way to generate mid-infrared ultrafast vortex beams and is of special significance for applications in the areas of molecular spectroscopy and organic material processing amongst others.

High-order femtosecond vortices converted from a tunable high-order Hermite–Gaussian Yb:CALGO laser

Femtosecond optical vortices open a variety of applications, ranging from femtosecond micro-processing to vortex strong-field physics. Thus far, the generation of high-order femtosecond vortices remains a challenge. The 1st to 4th order tunable femtosecond vortices generated by a z-type cavity semiconductor saturable absorber mirror mode-locked Yb:CALGO laser are reported in this study. The anti-reflection coated Yb:CALGO crystal is used as the gain material and is positioned perpendicular to the laser beam to avoid an angle dispersion, which results in multi-transverse-mode running. The astigmatism caused by the tilted curved mirrors is compensated by elevating the folding mirrors adjacent to the curved mirrors. The 1st to 5th order tunable femtosecond Hermite–Gaussian (HGn0) modes are realized by employing the off-axis pumping technique. The mode-locked pulse width is in the range of 220–580 fs for various HG modes. The mode-locked HG modes are converted into femtosecond vortices using a cylindrical-lens mode converter. The work provides a reliable way to generate high-order femtosecond vortex beams.

Index-Tunable Structured-Light Beams from a Laser with an Intracavity Astigmatic Mode Converter

Structured-light lasers are highly topical due to not only their ability to tailor customized distribution of intensity, phase, orbital angular momentum (OAM), and other optical properties, but also the compact and simple at-the-source generation scheme. Here we propose a form of structured-light laser, intracavity mode convertor laser, with the ability to generate two-dimensional (2D) tunable high-order Hermite-Gaussian (HG) modes with a large indice range (up to 15). The mode convertor constituted by two cylindrical lenses is always used for external OAM conversion, but here we show that it can be inserted into a laser cavity as an intracavity element for symmetry-breaking control, which we also demonstrate with a complete matrix optics theory. The generated 2D HG modes can also be converted into Laguerre-Gaussian (LG) vortex beams with both tunable OAM and radial momentum. Moreover, we also show the ability to directly generate vortex beam from our laser cavity. Our approach meets the urgent necessity for a practical laser device for more versatile light, providing more alternative parametric space for structure control and fostering extended applications.

High-Precision Multiparameter Weak Measurement with Hermite-Gaussian Pointer

The weak-value amplification technique has been proved to be useful for precision metrology in both theory and experiment. To explore the ultimate performance of weak-value amplification for multiparameter estimation, we investigate a general weak-measurement formalism with the assistance of the high-order Hermite-Gaussian (H-G) pointer and the quantum Fisher information matrix. Theoretical analysis shows that the ultimate precision of our scheme is improved by a factor of the square root of $2n+1$, where $n$ is the order of the Hermite-Gaussian mode. Moreover, the parameters' estimation precision can approach the precision limit with the maximum-likelihood estimation method and the homodyne method. We also present a proof-of-principle experimental setup to validate the H-G--pointer theory and explore its potential applications in precision metrology.

Topological spinor vortex matter on spherical surface induced by non-Abelian spin-orbital-angular-momentum coupling

We propose a scheme to implement non-Abelian spin-orbital-angular-momentum (SOAM) coupling in spinor Bose–Einstein condensates using magnetic gradient coupling. For a spherical surface trap addressable using high-order Hermite–Gaussian beams, we show that this system supports various degenerate ground states carrying different total angular momenta J, and the degeneracy can be tuned by changing the strength of SOAM coupling. For spinor condensates with hyperfine spin F = 1, the system supports various meta-ferromagnetic phases and meta-polar states always described by quantized total mean angular momentum in case of weak interactions. Polar states with Z2 symmetry and Thomson lattices formed by defects of spin vortices are also discussed. The system can be used to prepare various stable spin vortex states with nontrivial topology, and serve as a platform to investigate strong-correlated physics of neutral atoms with tunable ground-state degeneracy.

Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter.

The generalized geometric mode with several high-order Hermite-Gaussian (HG) beams localized on ray periodic orbits in the degenerate resonator is generated by an off-axis pumped Nd:YVO4 laser, by performing beam transformation via an astigmatic mode converter, the generalized geometric modes are found that can be converted into the multi-axis vortex beams with the bundled-rings structures. Experimental results reveal that the generated multi-axis vortex beams can preserve quite stable beam structures even under high-power operation. Moreover, the radius of the bundled rings for the multi-axis vortex beams can be flexibly adjusted by the off-axis pumping to lead to vortex structures with easily controlled orbital angular momentum distribution. The good agreement between the experimental and theoretical results of propagation evolution for the astigmatic transformation of generalized geometric modes further verify the feasibility of using the proposed system to realize various high-powered, multi-center vortex beams with good reliability and predictability.

Wavelength-tunable Hermite-Gaussian modes and an orbital-angular-momentum-tunable vortex beam in a dual-off-axis pumped Yb:CALGO laser.

A dual-off-axis pumping scheme is presented to generate wavelength-tunable high-order Hermite-Gaussian (HG) modes in Yb:CaGdAlO4 lasers. The mode and wavelength can be actively controlled by the off-axis displacements and pump power. The purities of the output HG modes are quantified by intensity distributions and the measured M2 values. The highest order reaches m=15 for stable HGm,0 mode, and wavelength-tunable width is about 10nm. Moreover, through externally converting the HGm,0 modes, the vortex beams carrying orbital angular momentum (OAM) with a large OAM-tunable range from ±1ℏ to ±15ℏ are produced. This work is effective for largely scaling the spectral and OAM tunable ranges of optical vortex beams.

Characterizing the propagation evolution of wave patterns and vortex structures in astigmatic transformations of Hermite–Gaussian beams

Theoretical wave functions are analytically derived to characterize the propagation evolution of the Hermite–Gaussian (HG) beams transformed by a single-lens astigmatic mode converter with arbitrary angle. The derived wave functions are related to the combination of the rotation transform and the antisymmetric fractional Fourier transform. The derived formula is systematically validated by using an off-axis diode-pumped solid-state laser to generate various high-order HG beams for mode conversions. In addition to validation, the creation and evolution of vortex structures in the transformed HG beams are numerically manifested. The present theoretical analyses can be used not only to characterize the evolution of the transformed beams but to design the optical vortex beams with various forms.

Generating high-peak-power structured lights in selectively pumped passively Q-switched lasers with astigmatic mode transformations

Various high-order Hermite–Gaussian (HG) modes with high repetition rates and high peak powers are systematically generated by designing the cavity configuration to satisfy the criterion of the passive Q-switching. For the HGm,0 modes with the order m = 1–9, the pulse repetition rate can exceed 100 kHz with peak power higher than 0.3 kW. For the HGm,m modes with the order m = 1–10, the pulse repetition rate can be up to 37 kHz with peak power higher than 0.35 kW. Furthermore, the high-order HG beams is transformed by using an astigmatic mode converter to generate various structured lights with optical vortices. Experimental patterns of the transformed high-order HG beams in the propagation are theoretically analyzed and the phase structures are numerically manifested.

Generation of picosecond vortex beam in a self-mode-locked Nd:YVO4 laser

In this paper, a self-mode-locked Nd:YVO4 picosecond vortex laser is demonstrated, which can operate on the different Laguerre-Gaussian (LG) modes at 1 064 nm. A π/2 mode converter is utilized to realize the picosecond vortex laser with LG mode transformed from the high-order Hermite-Gaussian (HG) mode. For the proposed laser, the mode-locked pulse repetition rate is 1.81 GHz. The average output powers of LG12 mode and LG02 mode are 1.241 W and 1.27 W, respectively, and their slope efficiencies are 23.2% and 24%, respectively.

Generation of Higher Order Vortex Beams From a YVO<sub>4</sub>/Nd:YVO<sub>4</sub> Self-Raman Laser via Off-Axis Pumping With Mode Converter

The generation of high-order Hermite-Gaussian (HG) beams at the Stokes wavelength in a YVO4 /Nd:YVO4 self- Raman laser with off-axis pumping is demonstrated. The high- order vortex beams at the Stokes field are successfully created by transforming these HG modes via an extra-cavity mode converter. The stimulated Raman scattering (SRS) threshold pump power for the diode-end-pumped laser is theoretically analyzed to verify the feasibility of generating high-order HG modes by off-axis pump- ing. At a pump power of 18.6 W, the SRS output powers can be higher than 1.0 W for the HG modes from TEM0 ,0 to TEM3 ,0 . Under the same pump power, the maximum order can be up to TEM 28 ,0 . Index Terms—Optical vortices, Raman scattering, diode pump- ing, Beam transformation.