渦旋光激光器研究進展（特邀）

呂銘鑫，張翼鵬，何健朗，胡小鵬，張勇

（1 固體微結(jié)構(gòu)物理國家重點實驗室（南京大學(xué)），南京210093）

（2 南京大學(xué)現(xiàn)代工程與應(yīng)用科學(xué)學(xué)院，南京210093）

（3 南京大學(xué)物理學(xué)院，南京210093）

0 Introduction

The laser，as a light source with excellent monochromaticity，coherence，directionality and energy density，has been widely used in industry，agriculture，communications and aerospace since its invention.The history of its development can be traced back to the concepts of stimulated absorption and stimulated radiation［1］.Recently，there has been increasing demand for a vortex beam that has a spiral wavefront，i.e.，a helical phase with a period of 2πl(wèi).Here，lis the topological charge.In 1992，ALLEN L et al revealed the relationship between the spiral wavefront and Orbital Angular Momentum（OAM）［2］，i.e.，each photon of vortex beam carries an OAM ofl?（where ? is the reduced Planck constant）.In addition to the amplitude，phase，polarization and propagation properties，the OAM is also considered as an essential parameter for a vortex beam.These unique properties boost the applications of vortex beam in micromachining，optical communications，optical manipulation，superresolution imaging，precision measurement，nonlinear and quantum optical applications［3-14］.The vortex beam laser becomes a key device that outputs a high-quality vortex beam.Currently，the researches on vortex beam laser have been focused on realizing high efficiency，high mode purity，and system integration.

In this paper，we will introduce the recent developments of vortex beam laser.First，we compare the vortex beams generated by using active and passive methods.Then，the research progresses on vortex beam laser，including solid-state laser，optical parametric oscillator，fiber laser and on-chip integrated laser，are reviewed.Finally，we discuss the prospects of the future vortex beam laser.

1 Vortex beams

One typical vortex beam is Laguerre-Gaussian（LG）beam，which is the eigen solution of the scalar Helmholtz equation，?2u+k2u=0，in cylindrical coordinates.It can be expressed as

where（r，φ，z）denotes the cylindrical coordinates andkis the wave vector.l（taking any integer value）is the azimuthal index of the LG beam andp（taking a zero or positive integer value）is the radial index.LG beam features a helical phase of exp（ilφ）and a phase singularity at the center.Other types of vortex beam include high-order Bessel beams，hypergeometric Gaussian modes and helical-Ince-Gaussian modes［15-17］.In 2020，SHEN Y J et al proposed a new type of vectorial vortex beam，which is described as the superposition of SU（2）coherent states［18］.Such vectorially structured light features controllable degrees of freedoms in terms of coherent-state phase，OAM，polarization and trajectory shape，surpassing the limit of the traditional vortex beam（Fig.1）.

Fig.1 Several generalized Poincaré sphere of SU（2）coherent state［18］

The methods of generating vortex beam can be divided into two categories，i.e.，passive and active methods.The passive method generally loads the spiral phase into a Gaussian beam by using phase modulation elements such as forked gratings，Spiral Phase Plate（SPP），Vortex half-Wave Plate（VWP）andq-plate［19-24］.However，the vortex beam generated in these methods is generally a hypergeometric Gaussian（HyGGl）mode，which is composed of multiple LG modes with the samelbut variousAs shown in Fig.2（a），if we use a passive method to generatel=1 mode，only 78% of the energy is concentrated on LG1，0mode.The situation becomes worse whenlincreases.This will lead to severe problems when the applications require a high-purity LG mode［25-27］.For example，high-order LG beam can be used in Laser Interferometer Gravitational-wave Observatory（LIGO）system to detect gravitational wave［13］.As shown in Fig.2（b），the thermal deformation of mirror surface can be significantly suppressed by using an LG mode with highlandpindices.In this case，high mode purity is the key to enhance the signal-to-noise ratio.Therefore，in order to improve the mode purity，it is necessary to filter out the unwanted modes，which generally leads to complex system and low conversion efficiency.Under such circumstances，the active method of vortex beam laser is viewed as a promising way to obtain a high-quality vortex beam.Next，we focus on the recent development of vortex beam lasers.

Fig.2 The necessity of producing high purity vortex beam using active methods

2 Laser output of a vortex beam

Unlike the passive generation of a vortex beam，the active way uses the properties of a laser resonant cavity to directly output a vortex beam.It is critical to carefully design the laser cavity，such as controlling the gain and loss of the cavity modes，matching the self-reproduction condition in cavity，and so on，to ensure the purity of the desired vortex beam mode.The following will be introduced in detail from the aspects of solid-state lasers，optical parametric oscillations，fiber lasers and integrated lasers.

2.1 Solid-state vortex lasers

The solid-state vortex laser was first developed.In principle，one can obtain different laser output modes through proper design of the resonator.For a resonant cavity，it is also possible to output multiple different eigen modes at the same time.In general，one can effectively control one or more modes to output by manipulating the gain and loss of various modes in the cavity.As an example，when one need to output a Gaussian mode，it is effective to suppress other higher-order cavity modes.Similarly，if we need a vortex beam to output，other modes including the Gaussian mode have to be suppressed［28-31］.

Pump shaping is an effective way to accomplish this task.When shaping the pump into a ring-shaped beam，the vortex beam mode is effectively amplified while the Gaussian mode is not.In 2005，BISSON J F et al realized a high-order vortex beam through hollow-shape pumping［29］.They input a pulsed laser beam having a wavelengthλ=0.532 μm on a circular diaphragm to produce a hollow intensity distribution in the near field.Then，they used this beam to pump an Nd：YAG laser and obtained the laser output of LG modes，the order of which ranged from 1 to more than 200.In 2015，KIM D J et al obtained the output of two degenerated LG beams with opposite values oflusing a ring-shaped pumping beam［30］.To obtain a pure LG beam，they used a tilted etalon inside the resonator，which broke the propagation symmetry of the Poynting vectors with opposite helicity（Fig.3）.

Fig.3 The experimental setup with a tilted etalon for laser output of an LG beam［30］

A vortex laser can be achieved by introducing an additional loss of the Gaussian mode in a resonant cavity［32-35］.In 2010，ITO A et al generated both scalar and vector hollow beams using a side-pumped Nd：YAG laser cavity［32］.To suppress the oscillation of lower-order transverse modes，they used a mirror with a lowreflectivity central spot.In experiment，they generated a hollow scalar beam（LG beam）and vector beams（LG and Bessel-Gaussian beams）.In 2011，NAIDOO D et al output a vortex beam by placing a stop in front of the output coupler in a cavity［33］.In this way，they generated a coherent superposition of two LG modes yielding“petal”modes in a standard Fabry-Pérot laser cavity.The azimuthal mode order was up to 8 by adjusting the stop.In 2010，THIRUGNANASAMBANDAM M P et al used a simple short-focus plano-convex glass lens with strong spherical aberration for Laguerre-Gaussian mode selection in a Continuous Wave（CW）laserdiode-end-pumped Yb：YAG ceramic laser［36］.They observed a sequence of LG modes with different combinations of radial and azimuthal indices（p=0～12，l=0～28）.The output power was up to 30 mW at a wavelength of 1 030 nm.

One recently-developed method to generate a vortex beam is using spiral phase modulation elements in a cavity［36-42］.The advantage of this method mainly lies in high purity of the generated vortex beam.After the phase modulation element is added，the condition of the self-reproduction of the mode in the cavity must be properly satisfied to allow the vortex beam output.Here，we use VWP as an example to demonstrate the conversion process.VWP can be regarded as a half-wave plate whose direction of the fast axis changes with the azimuthal angle.Its Jones matrix can be expressed as

whereαis the orientation angle of the fast axis of the VWP.The relationship betweenαand the azimuthal angleθis

which is dependent on the OAM orderland the initial azimuthal angle of the fast axisα0.For a Left Circularly Polarized（LCP）incident light，the transmitted light is

where and are the Jones vectors of the LCP and Right Circularly Polarized（RCP）lights，respectively.Eq.（4）shows that the VWP can transform LCP light into RCP light carrying a spiral phase term of e-ilθ，and vice versa.A vortex beam can thus be obtained using circularly polarized light and a VWP.In 2016，NAIDOO D et al showed that any Higher-Order Poincaré（HOP）sphere beam can be directly generated by using intracavity phase modulation elements［41］.By controlling the relative angle between a quarter wave plate and aq-plate，they adjusted the geometric phase distribution in the output light to produce any arbitrary beam on the HOP sphere，including Cylindrical Vector（CV）vortex beams（such as azimuthally and radially polarized lights）and OAM modes.The mode purity was as high as ＞98%.In 2019，WEI D Z et al output the LG mode using reversible cavity mode conversion and appropriate cavity design［42］.By using a VWP，a Faraday Rotator（FR）and a Quarter-Wave Plate（QWP）in a Nd：YVO4laser cavity，they realized the self-reproduction of the cavity mode and obtained low lasing thresholds（0.7 W forl=1 and 1.8 W forl=2），high slope efficiencies（11.06% forl=1 and 5.11% forl=2）and high mode purities（97% forl=1 and 93% forl=2）of LG mode lasers（Fig.4）.

Fig.4 The experiment of LG beam laser with a VWP［42］

The use of SLM in a resonator makes it possible to produce different structured light fields through the loading of different holograms.In 2013，NGCOBO S et al used a phase-only reflective SLM to form a rewritable holographic mirror in place of the standard laser cavity mirror［40］.In principle，any light field can be output by loading the corresponding holograms on the SLM.In experiment，they generated HG modes，LG modes，and flat-top modes of different orders.

Recently，metasurface is utilized in a laser cavity for vortex beam generation.The unique advantage of this arrangement is that metasurface can be feasibly designed to realize specific phase modulations according to need.In 2020，SROOR H et al designed custom metasurfaces for arbitrary OAM coupling to linear polarization states，including one with an extreme helicity of up tol=100［43］.In this manner，they were also able to produce new chiral states of light from a laser，including simultaneous lasing across vastly differing and non-symmetric OAM values that are up to Δl=90 apart.

2.2 Vortex-beam optical parametric oscillator

Compared with solid-state lasers， the Optical Parametric Oscillator （OPO） produces different wavelengths of output light depending on the phase-matching conditions.By changing the temperature，cut angle，or the period of quasi-phase-matching crystal，OPO is able to generate a continuously-tunable wavelength range of the output light.

In 2004，MARTINELLI M et al investigated the transfer of OAM in an OPO system［44］.In experiment，they used a 10-mm-long KTP crystal，which was cut for noncritical phase matching at 1 064 nm.The OPO was pumped by an LG mode beam converted from HG mode beam using two cylindrical lenses.By controlling the depletion in cavity，they successfully achieved LG beam output in signal or idler mode（Fig.5）.

Fig.5 The experiment of the vortex-beam OPO［44］

In 2011，MIYAMOTO K et al demonstrated a mid-infrared OPO pumped by a 1-μm optical vortex，and they created fractional vortex pulses having half-integer topological charge［45］.The fractional vortex pulse had an energy of 0.5 mJ and a width of 31 ns.The generation of 0.24-mJ mid-infrared vortices with a topological charge of 1 was also achieved.The tuning range covered the wavelengths from red to mid-infrared（735 nm～12.5 μm）［46-50］.

In 2017，AADHI A et al realized a vortex-pumped continuous-wave OPO［51］.They used a doubly resonant oscillator and quasi-phase-matching nonlinear crystal to achieve a high gain and avoid anisotropy effects due to birefringence.They theoretically and experimentally verified that by controlling cavity losses，one can selectively transfer the OAM carried in the pump light to either the signal or idler light.The experimental setup and output beam patterns are shown in Fig.6.They showed that when pumping with the OAM states |lp|=1 and |lp|=2，the OPO has the two output states（|1，0＞and |0，1＞）and three output states（|2，0＞，|1，1＞，and |0，2＞），respectively.

Fig.6 The experiment of OAM-mode-pumped doubly-resonant OPO［51］

Because of the low energy density of the vortex pumping light，the frequency conversion efficiencies in above OPO configurations are generally low.In 2020，WEI D Z et al proposed and experimentally realize highlydemonstrated a Janus OPO system by introducing an intracavity mode conversion system，which is capable to efficient output of highly pure，broadly tunable and topological-charge-controllable LG modes［52］.A speciallydesigned imaging system was introduced into the cavity to guarantee the smooth conversion between the Gaussian mode and LG mode.The output LG mode had a tunable wavelength between 1.5 μm and 1.6 μm with a conversion efficiency above 15%，a topological charge switchable from?4 to 4，and a mode purity as high as 97%.

2.3 Fiber vortex lasers

A fiber laser typically uses glass fibers doped with rare earth elements as gain mediums.The fiber laser has good heat dissipation and excellent beam quality.Moreover，a fiber laser system is generally low-cost，which has undergone rapid development in recent years.The investigation of the fiber vortex beam laser has received increasing attentions for its application potentials in vortex-beam fiber optic communication［53-58］.

In 2016，ZHANG W D et al generated a vortex beam using an acoustically induced fiber grating driven by a radio frequency signal，which converted the left/right-handed circular-polarization fundamental mode to the+1/?1 order optical vortex in a two-mode fiber［55］.The mode conversion efficiency was kept at ～95% with a wavelength range of 1 540～1 560 nm by tuning the frequency of the radio frequency driving signal.

In 2017，WANG T et al generated high-order pulse vortex beams using a mode-locked fiber laser［56］.They fused a mode-selective coupler using a Single-Mode Fiber（SMF）and a Few-Mode Fiber（FMF）and converted the LP01mode to a broadband LP11（LP21）mode.The measured durations of the vortex beam pulses were 273 fs and 140 fs for OAM±1and OAM±2modes，respectively.The maximal single-pulse energy was 0.36 nJ，which can be improved by increasing the pump power or reducing the loss of the SMF-FMF coupler.

In 2021，SHA W et al used a polarization rotation technique to select the fiber laser output mode［57］.By rotating waveplates inside the cavity，losses of different modes can be modulate.LP01mode，LP11mode，LP21mode and vortex beams with topological charges of±1 can be obtained from the fiber laser cavities（Fig.7）.

Fig.7 The experiment of fiber vortex laser with polarization rotation technique［57］

2.4 On-chip integrated vortex lasers

Along with the rapid development of microfabrication techniques，the on-chip integrated vortex lasers has become a hot topic of research［59-65］.In 2001，CHEN Y F et al fabricated a double-end-pumped microchip laser to explore the output law of LG mode［59］.They found that two LG modes can be directly superposed without appreciable overlapping between their excited regions.In contrast，strong competition occurs when the inversion populations have considerable overlap.Later，they investigated the dynamics of transverse patterns in solidstate microchip lasers with a large Fresnel number.By controlling the transverse-mode spacing and the mode size，they generated a stable transverse pattern of optical vortex lattices.

In 2016，MIAO P et al realized the single-mode output of OAM microlaser［60］.They used a microring cavity supporting whispering gallery modes，which can output laser with a large OAM.In general，one needs to add a nonreciprocal isolator to break the reciprocity between the counter propagating waves in the microring cavity.By introducing complex-refractive-index modulations to form an exceptional point，the unidirectional propagation of waves was promoted.The topological charge and vector polarization states can be controlled by properly designing the resonator.

In 2019，ZHANG Z L et al directly generated a vortex beam from polarization mode selection in a dualpolarization microchip laser，which realized the simultaneous existence of TEM00mode and LG mode with different frequencies and polarizations in the cavity［61］.Because of the frequency difference and spatial holeburning effect，mode competition leaded to the generation of the x-polarized LG01mode and y-polarized TEM00mode，which can be easily separated with a polarizing beam splitter.

3 Conclusion

The vortex beam laser has experienced rapid development in recent years.Its applications have been extended from the traditional optical micromachining and optical manipulation to superresolution imaging，precision measurement， optical communications and optical information processing.These high-end applications significantly boost the requirements of high-quality vortex beam from various types of lasers.The development of high-performance vortex beam laser can also stimulate the research interests of fundamental physics involving the OAM of light，such as the manipulation of OAM in a nonlinear optical process，the OAM-enhanced capacity in quantum communications，and so on.

In future，there still remains many challenges for the vortex beam laser.As an example，it is critical to improve the output efficiency and power of the vortex beam laser.For certain extreme applications such as interferometric measurements in LIGO system，it requires high purity and high power output of high-order LG modes.In addition，the on-chip integrated vortex laser has attracted increasing research interests.It is exciting to anticipate a miniature laser device that output high-performance vortex beam.