Vortex arrays directly generated from an efficient diode-pumped microchip laser

Optical vortex arrays with multiple singularities arranged in distinct lattice structures provide more flexibility in trapping or manipulating microparticles, large-capacity optical communications and high-security information processing, and optical modulation. An efficient high-power compact laser with singularity-tunable vortex-array distribution is crucial for practical applications. Here, we directly generate various vortex arrays with tunable singularities from 1 to 10 in a microchip laser pumped with a tilted annular beam. Formation of vortex arrays in the microchip laser is achieved by manipulating the gain distribution in an Yb:YAG crystal by controlling the tilt angles and pump power. Efficient high-power laser operation with output power of 2.01 W and optical efficiency of 24.5% is obtained for a vortex array with 10 singularities. The pump-power-dependent saturated population inversion distribution plays a key role in generating vortex arrays formed with multi-transverse modes oscillating simultaneously in transverse-mode locking. The good agreement between theoretically simulated transverse patterns, phases and interference patterns for the vortex arrays and experimentally obtained results provides a solid foundation for developing efficient compact microchip lasers for generating vortex arrays by manipulating gain distribution in a thin medium pumped with a tilted annular beam.


Introduction
Vortex lasers carrying orbital angular momentum vertical-cavity surface-emitting [1] have been widely used in optical trapping [2], quantum communication [3], material processing [4], high -capacity storage [5], information processing [6], quantum entanglement [7], and high -resolution imaging [8]. Optical vortex arrays or vortex lattices [9][10][11][12] with multiple vortices arrang ed in a spatial distribution provide more flexible applications for trapping or manipulating microparticles [13][14][15], large -capacity optical communications and high -security information processing [16], optical modulation [17], etc. Therefore, various methods have been proposed for developing optical vortex arrays. Complex transverse patterns with spatially distributed vortices have been widely investigated theoretically and experimentally [18] since the concept of phase -singularity crystals was proposed [19]. Various phase -singularity crystals have been investigated theoretically based on the transverse-mode -locking effect in a frequency-degenerated family of modes, and demonstrated experimentally in an argon-ion laser pumped Na 2 molecular laser [19]. Various optical vort ex crystals have been generated spontaneously in a semiconductor vertical-cavity surface-emitting laser [20]. However, the complexity of a vapor laser and low beam quality of a semiconductor laser have limited the applications of vortex arrays. Alternatively, optical vortices and vortex arrays have been generated by passing a single-mode laser through a specially designed spatial light modulator [7,17,[21][22][23], spiral phase plate [24], and hologram formed with liquid crystal [25]. However, the beam quality is degraded and transverse patterns are distorted because it is hard to achieve a highly precise process of the phase elements [26]. Diffraction loss and low damage threshold of these optical phase components also limit high power operation. Recently, various vortex-array beams have been generated in microchip lasers [9,12] and thin-slice solid-state lasers [27,28]. Square vortex arrays have been generated in a solid-state microchip laser with a The schematic of generating vortex arrays in a tilted annular-shaped fiber-coupled laser-diode -pumped microchip laser is shown in figure 1. The annular-shaped pump source and optics coupling system are the same as that used for generating high-order cylindrical vector beams [37]. The tilted annular-shaped laser-diode -pumped Yb:YAG microchip laser was realized by tilting the microchip laser head 5 • with respect to the normal incident plane of the pump beam. The gain distribution is manipulated by focusing the annular beam 0.3 mm in an Yb:YAG crystal [41]. A 1 mm thick, coated Yb:YAG crystal doped with 10 at.% Yb 3+ ions and a plane-parallel output coupler (OC) with 95% reflection were attached tightly to form microchip cavity. High reflection (HR) at 1030 nm and anti-reflection (AR) at 940 nm were coated on one surface of the Yb:YAG crystal to act as the rear cavity mirror (M1). The other surface of the Yb:YAG crystal was coated with AR at lasing wavelength to reduce diffraction and transmission losses. A beam profile analyzer (Thorlabs BC106-VIS) was employed for measuring the intensity profiles of the vortex-array beams. An Anritsu optical spectral analyzer (MS9740A) was used to measure the laser spectra of different vortex arrays.

Results
The lasers were examined under continuous-wave (CW) condition at input pump power (P in ) from 1 W to 8.2 W. The Yb:YAG microchip laser oscillated when P in reached 1.7 W. The near-field intensity patterns generated experimentally at different P in s were measured by transmitting the laser through a focus lens (100 mm focal length) on a beam profile analyzer. Within the available pump power range up to 8.2 W, 11 typical transverse patterns were obtained by adjusting the pump power. The intensity profiles remained as a Gaussian distribution when P in increased from the threshold pump power to 2 W. One typical transverse pattern with Gaussian distribution at P in = 1.8 W is shown in figure 2(a1). When P in was increased over 2 W, the intensity profile changed from a Gaussian distribution to a doughnut. The pump power range for doughnut-shaped laser oscillation was from 2 W to 3.1 W. One typical doughnut-shaped laser profile at P in = 2.5 W is shown in figure 2(b1). The experimentally obtained transverse patterns with Gaussian Figure 1. Experimental setup of an annular-shaped laser-diode-pumped Yb:YAG microchip laser for directly generating singularity-tunable vortex arrays. f 1 and f 2 are the collimating and focus lenses with a focal length of 8 mm. OC is the output coupler. θ is the tilt angle of the Yb:YAG crystal with respect to the annular pump beam. The inset shows the measured annular intensity profile from a fiber-coupled laser diode. distribution and doughnut profile, as shown in figures 2(a1) and (b1), are in good agreement with the theoretical simulated TEM 00 mode and LG 0,1 mode, as shown in figures 2(a2) and (b2).
Further increas ing P in over 3.1 W, the near-field transverse profile changed from a doughnut pattern to complex transverse patterns characterized by arrays of vortices with null intensities, namely, vortex-arrays. Figure 3 shows the typical vortex-arrays experimentally generated in the Yb:YAG microchip laser at different P in s, together with the schematic distribution of the vortices. A two-vortex array consisting of two separated vortices was observed when P in was higher than 3.1 W ; one example at P in = 3.5 W is shown in figure 3(a). A three-vortex array consisting of three vortices arranged along a line was obtained when P in reached 4 W. One example of a three-vortex array at P in = 4.2 W is shown in figure 3(b). A four-vortex array with four vortices in a diamond shape was obtained when P in increased to 4.6 W. One typical four-vortex array obtained experimentally at P in = 4.8 W is shown in figure 3(c). When P in reached over 5.1 W, the laser oscillated in a five-vortex array consisting of five vortices arrang ed as a vortex surrounded by four vortices in a rectang ular structure. A typical five-vortex array at P in = 5.2 W is shown in figure 3(d). A six-vortex array consisting of six vortices in a doubled diamond structure was generated when P in was higher than 5.6 W. A typical six-vortex array at P in = 6 W is shown in figure 3(e). A seven-vortex array consisting of seven vortices arrang ed as a vortex centered in a distorted hexagon was obtained when P in was higher than 6.4 W. A typical seven-vortex array at P in = 6.5 W is shown in figure 3(f). An eight-vortex array consisting of eight vortices arrang ed in a lattice with two vortices between two lines of three vortices was observed when P in was higher than 6.9 W. One example of an eight-vortex array at P in = 7 W is shown in figure 3(g). A nine-vortex-array consisting of nine vortices in a lattice structure was obtained when P in was higher than 7.3 W. One typical nine-vortex array at P in = 7.5 W is shown in figure 3(h). A ten-vortex-array consisting of ten vortices arrang ed as two separated vortices surrounded by an octagon was achieved as P in reached 8.1 W. One typical ten-vortex-array at P in = 8.2 W is shown in figure 3(i).
The threshold pump powers (P th ) for different vortex arrays were determined by slowly adjusting P in for observing the transition from one transverse pattern to another. The transition from one transverse pattern to another is sudden and abrupt owing to the pump -power -dependent transverse mode competition. However, the observed transverse patterns are stable once the transverse patterns are generated. Figure 4 depicts the variation of P th s for different vortex arrays with vort ex number (m), together with the P th s for TEM 00 and doughnut modes. The P th for TEM 00 mode was 1.7 W, and the P th for doughnut mode was 2 W. A near -linear increase of P th with vort ex number was observed for the vortex arrays. The increased rate of P th is about 0.6 W per vort ex number. Wide separation of the threshold pump powers between two adjacent  vortex arrays is more favorable for maintaining stable operation of a specific vortex-array beam generated in the compact microchip laser. Vortex -number -controllable vortex arrays with wide pump power range provide more flexib ility for various applications and are favorable for developing a compact laser.
The phase helicities of the vortex arrays obtained in the microchip laser w ere verified with interference experiments. A plane-wave reference beam was formed by spatially selecting a tiny portion of a vortex-array beam, and then interfer ing with the vortex-array beams to generate interference patterns. The interference pattern was captured with a beam profile analyzer. Figure 5 shows the experimentally obtained interference patterns of some vortex arrays emitted by the microchip laser (doughnut, two-vortex array, three-vortex array, four-vortex array, five-vortex array, six-vortex array). As shown in figure 5(a), the bifurcated forklike interference pattern clearly confirms that the doughnut beam is an optical vortex beam processing a phase singularity. The topological charge is ±1 depending on the helicity of the vortex beam. For the two-vortex-array beam, as shown in figure 5(b), two bifurcated forklike interference fringes verify that the two-vortex-array beam has two phase singularities. Each vortex has unitary topological charge. However, the phase helicit ies of the two vortices are opposite, thus the total charge of the two-vortex beam is 0. Similarly, the three-vortex-array beam processes three phase singularities each of unitary charge, and the total charge is  ±1, as shown in figure 5(c). The four-vortex array processes four singularities, and the total charge is 0, as shown in figure 5(d). As shown in figures 5(e) and (f), the measured interference patterns of the five-vortex array and six-vortex array clearly show that they possess five singularities and six singularities, each of unitary charge, and the total charges are ±1 and 0, respectively. Therefore, for the vortex arrays with even phase singularities such as 2, 4, 6, the total charge is 0; while for the vortex arrays with odd phase singularities such as 1, 3, 5, the total charge is ±1. The same phenomena were observed for the vortex arrays with vortex number larger than 6 generated in the microchip laser. The interference results further confirmed that the experimentally obtained transverse patterns (figure 3) are vortex arrays with structured distributed vortices, and each vortex possesses unitary topological charge. Figure 6 depicts the relationship between the output power and P in for the microchip laser, together with the variation of optical efficiency with P in . The P th s for TEM 00 and doughnut modes are 1.7 W and 2 W, respectively. Although the transverse intensity profile of the microchip laser changed with P in (as shown in figure 3), the output power of the vortex-array laser increased linearly with P in when P in was higher than 3 W. The increase rate of output power with P in (slope efficiency) was approximately 37.4%. A maximum output power of 2.01 W was achieved at P in = 8.2 W. The corresponding optical efficiency of 24.5% was 40% higher than that obtained in high-order cylindrical vector beam lasers [37]. No output power degradation was observed within the available maximum pump power. This indicates that the output power could be scaled by applying high pump power. It is also possible to generat e vortex arrays with phase singularities larger than 10 because high -order LG modes are favorable for oscillation to form a vortex array with large vortex number at high pump power supplied from an annular pump beam. The highly efficient performance of the tilted annular-shaped laser-diode -pumped Yb:YAG microchip laser was mainly caused by the transverse-mode -locking effect. The population inversion distributed along the Yb:YAG crystal length excited by the tilted focused annular beam is fully utilized by different participating transverse modes ; therefore, stationary vortex arrays are formed with multiple transverse modes oscillating simultaneously in transverse-mode -locking state.
The measured laser spectra at different P in s show that multi-longitudinal-mode oscillation around 1030 nm is dominant in the Yb:YAG microchip laser owing to the broad emission spectrum of the Yb:YAG crystal at approximately 1.03 µm [42]. Figure 7 depicts some typical oscillation spectra at different P in s. From the oscillation spectrum shown in figure 7(a), we can see that seven longitudinal modes oscillate at P in = 2.1 W. The longitudinal modes increase to 8 when P in is higher than 4.5 W. The eight-longitudinal -mode oscillation is maintained with further increasing of P in , as shown in figure 7(b) at P in = 5 W and figure 7(c) at P in = 7.2 W. The oscillation spectrum shifts to longer wavelength as P in increases, as shown in figure 7. The intensities of longitudinal modes also vary at different P in s because there is longitudinal mode competition. The separation between two consecutive longitudinal modes is 0.29 nm, which is in good agreement with the free spectral range (0.291 nm) of the microchip laser cavity filled with 1 mm Yb:YAG crystal. The frequency separation between the consecutive longitudinal modes is ∆ν L = c/2L, where c is the speed of light in a vacuum, and L is the resonator optical length. The frequency separation between the consecutive transverse modes is ∆ν [43], where p and l are the indices of a particular transverse mode, and R is the radius of curvature of the mirror. For the present plane-parallel microchip laser resonator, R is very large compared to L; therefore, the frequency separation between the transverse modes participating in forming a vortex array in the Yb:YAG microchip is small. ∆ν T is several orders of magnitude lower than ∆ν L . Thus, the resonant-mode spectrum is composed of the relatively large longitudinal mode spacing c/2 L, with each longitudinal mode surrounded by a set of transverse-mode resonances. The oscillation spectra with relatively stable multi-longitudinal-mode oscillations indicate that the transverse-mode -locking effect is responsible for forming stationary vortex-array transverse patterns in the Yb:YAG microchip laser pumped with a tilted annular beam. The multi-longitudinal-mode oscillation of the Yb:YAG microchip laser may cause some fluctuations in the laser output power and vortex arrays because the pulsation exists for each longitudinal mode [12]. It is possible to obtain stable oscillation in the Yb:YAG microchip laser due to the higher number of longitudinal modes extracting the energy for available manipulated gain distribution provided by the tilted annular pump beam.

Theoretical modeling
A microchip laser is a high-quality resonator filled with a high -nonlinearity gain medium. Its short cavity and large unrestricted lateral dimension enable a large Fresnel number (w p 2 /(λL c )), where w p is the effective pump beam radius, λ is the laser wavelength, and L c is the gain thickness. The Fresnel number of the microchip laser further increases with the effective pump beam diameter. A microchip laser with a large Fresnel number is favorable for high-order LG p,l mode oscillation. For a microchip laser pumped with a tilted annular beam, formation of vortex arrays is attributed to the asymmetrically distributed saturated population inversion inside the gain medium. The asymmetrically distributed saturated population inversion along the thickness and radial direction inside the gain medium provides possibilities for different LG p,l modes oscillating simultaneously. For a focused annular pump beam [37] tilted by an angle θ, the normalized pump power distribution can be expressed as follows [38] : where α and L c are the absorption coefficient and gain thickness, w p (z) is the position -dependent radius of the TEM 00 laser beam, r 0 = |z − z 0 |tan(θ p ) is the offset distance between the TEM 00 beam and propagation direction, 2θ p is the annular beam divergence angle, and z 0 is the position of the focus spot.
Owing to the tightly focusing properties of the annular beam, pump saturation effect at the focus spot has to be considered for estimating population inversion. Therefore, the P in -dependent saturated population inversion taking into account the pump saturation effect can be expressed as where I s is the saturated pump power intensity, and I p (r,z) is the pump power intensity. We defined the effective population inversion as the average distribution of the population inversion inside the gain medium. The effective population inversion distribution along the radial coordinate can be expressed as [38] Firstly, we theoretically investigate the effects of the annular beam tilting angle (θ) on the distribution of saturated population inversion inside the gain medium. Figure 8 shows the calculated saturated population inversion along the thickness and radial direction inside an Yb:YAG crystal at different θs. The parameters of the Yb:YAG crystal (α = 10 cm −1 , L c = 1 mm) and pump beam (w p0 = 20 µm, θ p = 10 • , z 0 = 0.3 mm) are used in the theoretical calculations. P in was set to 8 W in the theoretical calculation. When the incident angle of the annular beam is 0 • (normally incident on the Yb:YAG crystal), the saturated population inversion is distribute d symmetrically along the thickness (see figure 8(a1)). The effective population inversion is a ring shape, as shown in figure 8(a2). When the annular beam is tilted 2 • respective to the normal of the crystal surface, the saturated population inversion along the thickness is distorted (see figure 8(b1)). The distribution of the effective population inversion is changed from a symmetrical ring shape to a distorted ring shape with uneven population inversion distribution around the ring. As shown in figure 8(b2), there are two crescents with inversion population higher than 8 × 10 20 cm −3 for illustration. The saturated population inversion along the thickness of the Yb:YAG crystal is distorted with an increase in θ, as shown in figures 8(c1)-(e1). The effective population inversion distribution also tends to be further distorted with an increase in θ, as shown in figures 8(c2)-(e2). The highest effective population inversion increases with θ because the absorption of the pump power is proportional to the effective absorption length of the gain medium (L c /cosθ). The distribution of the effective population inversion further elongates along the y-axis and shrinks along the x-axis, as shown in figures 8 (c2)-(e2). Thus, the distribution of the effective population inversion excited in the Yb:YAG crystal can be manipulated by adjusting the tilt angle of the annular pump beam. Owing to the tightly focusing properties of the annular beam, the pump saturation effect around the focal spot further expands the distribution of the population inversion through the gain guiding effect. Therefore, the tilt -angle-dependent population inversion distribution provides an arena for multi-transverse modes oscillating simultaneously to form various stationary vortex arrays in the Yb:YAG microchip laser pumped with a tilted annular beam.
For the Yb:YAG microchip laser pumped with a tilted annular beam (as shown in figure 1), oscillation of TEM 00 , doughnut mode, and stationary vortex arrays are determined by the population inversion distribution in the Yb:YAG crystal, which relies on the applied pump power. Figure 9 shows the calculated population inversion along the thickness and radial direction in the Yb:YAG crystal at different P in s. The tilt angle of the annular beam, θ, was set to 5 • in the theoretical calculation. A red contour line (4 × 10 20 cm −3 ) and black contour line (8 × 10 20 cm −3 ) are plotted to indicate the change of the pump areas with P in , as shown in figures 9(a1)-(d1). The effective absorption length of the Yb:YAG crystal (population inversion > 4 × 10 20 cm −3 ) increases with P in . Besides the contribution to population inversion > 8 × 10 20 cm −3 coming from around the focus spot, the pumped area (population inversion > 8 × 10 20 cm −3 ) enlarges with an increase in P in . The effective population inversion distribution (figure 9(a2)) is an elongated ring with comparable population inversion, which is sufficient for supporting TEM 00 -mode oscillation. With further increasing P in , the pump saturation effect around the focus spot has a great effect on the distribution of the population inversion. The effective population inversion distribution tends to be an elongated ring along the y-axis with increasing P in , as shown in figures 9(b2)-(d2). Further increasing P in , the pumped area (population inversion > 8 × 10 20 cm −3 ) is enlarged (see figure 9(b1) at P in = 4 W). The effective population inversion distribution is an elongated ring with relatively low inversion population around the center, as shown in figure 9(b2). The population inversion increases with further increasing P in ; see figure 9(c1) at P in = 6 W and figure 9(d1) at P in = 8 W. The corresponding effective population inversion distribution exhibits an elongated ring with enlarged central area, as shown in figures 9(c2) and (d2). As shown in figure 9, the pump -power-dependent elongated ring distribution of effective population inversion provides an arena for simultaneous oscillation of different LG p,l modes in the Yb:YAG microchip laser pumped with a tilted annular beam. Various stationary vortex arrays are formed with transverse-mode -locking effect if the population inversion is sufficient to overcome the lasing thresholds of selected LG p,l modes. Therefore, it is reasonable in theory to generat e vortex arrays in the Yb:YAG microchip laser pumped with a tilted annular beam.
The oscillation of various LG modes depends on the mode coupling between the population inversion distribution and the distribution of LG modes. The transverse distribution of LG modes can be described with equations (5) and (6). The beam waist of the LG p,l modes is defined as the 1/e 2 intensity points of the outermost lobe, and increases with p and l. The oscillation of LG p,l modes with different beam waists and beam profiles depends on the gain distribution excited in the gain medium. Therefore, it is possible to form stationary transverse patterns with different transverse modes of oscillation simultaneously by manipulating the gain distribution inside the laser medium. Figure 10 depicts the profile distribution of the possible participating transverse modes for forming pump -power-dependent vortex arrays in the Yb:YAG microchip laser pumped with a tilted annular beam. The effective population inversion distribution is concentrated at low pump power, as shown in figure 9(a1), which is favorable for TEM 00 -mode laser oscillation ( figure 10(a)). Owing to the enhanced pump saturation effect with an increase in pump power, the effective population inversion at the center increases slowly compared to that around the ring ( figure 9(b2)). The gain for the TEM 00 mode decreases while the gain distribution is favorable for LG 0,1 -mode oscillation (see LG0,2 + LG1,2, (h) LG1,2 + LG2,0 + LG0,4, (i) LG0,1 + LG2,0 + LG0,4. The radii are normalized to the beam waist of the fundamental mode, w0. figure 10(b)) ; therefore, LG 0,l mode oscillates rather than TEM 00 mode. The effective population inversion distribution further elongates along the y-axis with further increasing pump power. The doughnut modes (LG 0,l , L G 0,−1 ) oscillate together with LG 1,0 mode owing to the increase in population inversion at the center with P in , as shown in figure 10(c). Therefore, a two-vortex array is formed with two opposite unitary phase singularities. With increas ing P in , the enhanced saturated population inversion distribution is favorable for oscillation of two orthogonal doughnut modes (LG 0,1 and LG 0,−1 ) together with two orthogonal LG 1,1 and LG 1,−1 modes, as shown in figure 10(d), thus a three-vortex array is formed. With further increase in P in , the effective population inversion distribution is expanded for high -order -mode oscillation. Therefore, the transverse modes such as LG 0,1 , LG 1,0 , LG 0,2 , LG 1,1 , LG 0,3 , LG 2,0 , LG 1,2 and LG 0,4 modes compete for the gain to form pump -power-dependent vortex arrays. As shown in figures 10(e)-(i), the formation of vortex arrays with certain transverse modes is determined by the pump -power-dependent saturated population inversion distribution in the Yb:YAG microchip laser pumped with a tilted annular beam.
The vortex arrays formed with transverse-mode -locking effect can be expressed with a linear combination of LG p,l modes with near frequency-degenerated mode locking. The electrical fields of LG p,l  −4(r,ϕ) modes, A p,l (r,ϕ), can be expressed as follows : where p = 0, 1, … is the radial index and l = 0, 1, … is the angular index, r is the radial coordinate, ϕ is the angular coordinate, w 0 is the beam waist of the TEM 00 mode, and L l p (x) is the Laguerre polynomials function. There are two forms for the electrical field of LG modes with l ̸ = 0. For simplicity, we use E p,l and E p,-l to express A pl1 and A pl2 . The electrical field of the vortex array is expresse d as a linear combination of LG p,l modes in the form E (r, ϕ) = g 1 E p1,0 (r, ϕ) + g 2 E p2,l2 (r, ϕ) e iθ2 + g 3 E p2,−l2 (r, ϕ) e iθ3 + g 4 E p3,l3 (r, ϕ) e iθ4 + g 5 E p3,−l3 (r, ϕ) e iθ5 (7) where θ 2 , θ 3 , θ 4 and θ 5 are the rotation angles of the possible transverse modes E p2,l2 , E p2,-l2 , E p3,l3 , and E p3,-l3 participating in form ation of stationary patterns, and θ 2 + θ 3 + θ 4 + θ 5 = π. g 1 , g 2 , g 3 , g 4 and g 5 are the weights of the transverse modes participating in formation of stationary transverse patterns. Therefore, stationary transverse patterns with different vortex distributions depend on the weights of participating modes, g 1, g 2, g 3 , g 4 , g 5 . The orientation of the transverse patterns of vortex arrays is determined by the particular values of θ 2 , θ 3 , θ 4 , and θ 5 .
Although the Yb:YAG microchip laser oscillates in multi-longitudinal modes at different P in s (figure 7), the stationary transverse patterns can be represented as a linear combination of transverse modes belonging to a frequency-degenerated family. Therefore, the stationary transverse patterns formed in the Yb:YAG microchip laser pumped with a tilted annular beam can be obtained theoretically with a set of rate equations with different modal coefficients [19,20]. The theoretically obtained transverse patterns oscillating in TEM 00 and doughnut modes are shown in figures 2(a2) and (b2), respectively. From figure 2, we can see that the experimentally obtained TEM 00 mode and doughnut mode are similar to the theoretically obtained patterns. The electrical field of the doughnut -mode beam can be expressed as E 0,1 (r,ϕ) or E 0,−1 (r,ϕ) depending on the helical phases. The transverse patterns of different vortex arrays were theoretically calculated by simultaneously oscillating different transverse modes within the mode-locking regime. The theoretically calculated transverse patterns with different vort ex distributions are shown in figure 11. The two-vortex array consisting of two vortices with two opposite helical phases is in simultaneous oscillation of LG 1,0 mode with two doughnut modes (LG 0,1 and LG 0,−1 ) with different weights and rotation angles. With further increase in the pump power intensity, the LG 1,1 mode is prior to oscillate other than LG 1,0 mode. Thus, the three-vortex array is formed with superposition of LG 1,1 and LG 1,−1 modes with doughnut modes (LG 0,1 , LG 0,−1 ). LG 1,0 mode and two doughnut modes (LG 0,2 and LG 0,−2 ) belonging to the 2p + l = 2 frequency-degenerated family are superposed to form four-vortex array [19]. The five-vortex, seven-vortex and nine-vortex arrays are formed with doughnut modes (LG 0,3 , LG 0,−3 ), LG 1,1 and LG 1,−1 modes belonging to the 2p + l = 3 frequency-degenerated family. The weights for the four modes depend on the gain profile provided with applied pump power from the tilted annular pump beam. The six-vortex array is a superposition of LG 0,2 , LG 0,−2 , LG 1,2 and LG 1,−2 modes. The eight-vortex array is a superposition of LG 2,0 , Figure 11. Theoretically calculated transverse patterns (a1)-(j1), phase (a2)-(j2) and interference patterns (a3)-(j3) of vortex arrays with tunable singularities.
LG 0,4 , LG 0,−4 , LG 1,2 and LG 1,−2 modes. The ten-vortex array is a superposition of LG 2,0 , LG 0,1 , LG 0,−1 , LG 0,4 and LG 0,−4 modes. The transverse modes participating in forming various vortex arrays with tunable phase singularities from 1 to 10 are listed in table 1. Figures 11(a2)-(j2) show the phase variation of the theoretically calculated vortex arrays. As shown in figure 11(a2), the phase variation of 2π counterclockwise around the vortex clearly confirms that the doughnut vortex has a phase singularity with unitary topological charge. For the two-vortex array, the phase variation of the two vortices is 2π ; however, the left one is clockwise varied while the right one is counterclockwise varied, as shown in figure 11(b2). For the three-vortex array, the phase variation of 2π is obtained for three vortices ; however, phase varies clockwise for the center vortex, while the other two vary counterclockwise, as shown in figure 11(c2). Similar phase variation of 2π is obtained for the vortices distributed in the vortex arrays with singularities larger than 3, as shown in figures 11(d2)-( j2). The calculated phases provide strong evidence for the vortices distributed in the vortex arrays being phase singularities with unitary topological charge. The theoretically calculated transverse patterns of vortex arrays with different vort ex distributions are in good agreement with the experimental results. The theoretically simulated results strongly support the mechanism of forming vortex arrays with multiple transverse modes oscillating simultaneously by fully using asymmetrical gain distribution in an Yb:YAG crystal pumped with a tilted annular beam.
The theoretically obtained transverse patterns of vortex arrays were used to interfere with a plane-wave reference beam. The calculated interference patterns for various vortex arrays are shown in figures 11(a3)-(j3). A bifurcated forklike interference pattern is obtained for each vortex distributed in the vortex arrays. The directions are opposite for the bifurcated forklike interference patterns of two vortices in the two-vortex array, as shown in figure 11(b3). The direction of one bifurcated forklike interference pattern is opposite to those of the other two for the three-vortex array, as shown in figure 11(c3). Thus, the total charge of the vortices is 0 for the vortex arrays with even vortices. The total charge is ±1 depending on the phase helicity for the vortex arrays with odd vortices. Good agreement is achieved between experimentally and theoretically calculated interference patterns for some vortex arrays ( figure 5). Therefore, the theoretically calculated interference patterns for the vortex arrays further confirm that formation of vortex arrays in an Yb:YAG microchip laser pumped with a tilted annular beam is attributed to the simultaneous oscillation of transverse modes in transverse-mode locking.

Conclusions
In summary, direct generation of regular vortex arrays has been demonstrated in an Yb:YAG microchip laser pumped with a tilted annular beam. The number of vortices with unitary topological charge increases linearly with pump power. An increase of pump power leads to the shift of oscillation spectra to lower frequencies. Efficient performance with output power of 2.01 W and optical efficiency of 24.5% has been achieved for a 10-singularity vortex-array laser. Formation of vortex arrays is attributed to multiple transverse -mode oscillation simultaneously supported with asymmetrical gain distribution provided by a tilted annular beam. The theoretically calculated transverse patterns, phases and interference patterns of the vortex arrays formed with linear combination of LG modes are in good agreement with experimentally obtained results. This work provides a flexible and robust method for developing singularity -tunable vortex arrays and structured light control [44] in microchip lasers. The vortex-array lasers with distinct singularity distribution are potential sources for effective trapping microparticles, high -capacity optics communication, and optical modulation.