This study reports the first systematic approach to the excitation of all high-order Hermite-Gaussian modes (HGMs) in end-pumped solid-state lasers. This study uses a metal-wire-inserted laser resonator accompanied with the “off axis pumping” approach. This study presents numerical analysis of the excitation of HGMs in end-pumped solid-state lasers and experimentally generated HGM patterns. This study also experimentally demonstrates the generation of an square vortex array laser beams by passing specific high-order HGMs (HGn,n + 1 or HGn + 1,n modes) through a Dove prism-embedded unbalanced Mach-Zehnder interferometer [Optics Express 16, 19934-19949]. The resulting square vortex array laser beams with embedded vortexes aligned in a square array can be applied to multi-spot dark optical traps in the future.
© 2012 OSA
Hermite-Gaussian modes (HGMs), Laguerre-Gaussian modes (LGMs), and the recently-observed Ince-Gaussian modes (IGMs) are three complete families of exact and orthogonal solutions to the paraxial wave equation [1, 2]. Chen et al  presented an approach to generating LGMs in end-pumped solid-state lasers (SSLs), while Thirugnanasambandam et al. found a way to generate very-high order LGMs in end-pumped Yb:YAG ceramic lasers . Researchers recently developed an approach to selectively generate a variety of higher-order IGMs from end-pumped SSLs by adjusting the pump position to one of the brightest spots (“target spot”) from which the stimulated emission builds up [5, 6]. As for the well-known HGMs, only HG0,n and HGn,0 modes can be directly generated from end-pumped SSLs with “off-axis pumping” scheme [7–10]; no other high-order HGMs have been observed in end-pumped SSLs. A variety of higher-order HGMs in gas lasers can be found in the literature [11, 12], where in the experiment a cross-hair is often inserted into the laser cavity with laterally uniform gain distribution. When the IGM ellipticity approaches infinity, the IGM approaches HGMs [1, 2]. The resulting IGM ellipticity can be modified slightly by tuning the pumping beam region . However, real experiments show that the “off-axis pumping” approach can only excite HG0,n or HGn,0 modes in end-pumped SSLs since the distinct target spot exists only for HG0,n or HGn,0 modes .
This study proposes a simple scheme for exciting all high-order HGMs from end-pumped SSLs with off-axis laser-diode (LD) pumping, where a fine opaque-wire is inserted into the laser resonator. This is the first systematic approach to exciting all high-order HGMs from end-pumped SSLs. This study verifies the HGM excitation method through both numerical calculations and experiments using a Nd:GdVO4 laser with laser-diode end pumping. It is noteworthy that a type of the resulting high-order HGMs, HGn,n modes, have potential application in multiple-trap laser tweezers [14–16]. This study also demonstrates a way to create a square vortex array laser beams by passing specific high-order HGMs (HGn,n + 1 or HGn + 1,n modes) through an unbalanced Mach-Zehnder interferometer . Optical vortex beams possess the special properties of carrying on optical angular momentum (OAM) and zero intensity behavior, and are now widely used in the transfer of angular momentum to micro particles or atoms [18–21]. The resulting square vortex array laser beams with embedded vortexes aligned in a square array can be applied to dark optical multiple-trap.
This paper is organized as follows. Section 2 describes the mechanism of exciting high-order HGMs in the proposed opaque-wire-inserted half-symmetric laser resonator. This section also addresses the generation mechanism, numerical exploration, and discussion of the resulting beam properties. Section 3 presents the experimental details and results of high-order HGMs generated from real end-pumped SSLs. Section 4 illustrates the generation of square vortex array laser beam by passing a specific high-order HGMs through an unbalanced Mach-Zehnder interferometer. This section analytically derives the vortex positions of the resulting square vortex array laser beams, and discusses its beam properties. Section 4 also presents experimental results of the generated square vortex array beams. Finally, Section 5 presents a short conclusion.
2. Selective excitation of high-order Hermite-Gaussian mode in end-pumped solid-state lasers with an opaque-wire-inserted laser cavity
2.1 Basic mechanisms: mode-gain control
This study uses a half-symmetric laser resonator and an “off-axis pumping” mechanism [5, 6] for the selective excitation of high-order HGMs. The resonator we used is similar to that used in real experiments [6, 22], but has an opaque-wire being inserted in the cavity. Figure 1(a) shows a schematic diagram of the opaque-wire-inserted laser resonator. The resonator consists of a planar mirror, a movable opaque-wire, and a concaved mirror at a distance from a planar mirror. The planar mirror is actually the high-reflection coated surface of a laser crystal. A movable opaque wire is inserted in the half-symmetric laser resonator. An objective lens focuses end-side pumping beam on the laser crystal. The transverse position and beam size of the end pumping beam can be easily controlled by modifying the objective lens.
The approach to excite HGMs in the end-pumped SSLs is based on the “mode-gain control” concept. This concept allows a desired HGM to be the mode that is most efficiently pumped in the cavity. The “mode-gain control” concept includes two points, i.e., gain control and loss control. The red spot in Fig. 1(b) illustrates the transverse position of the pumping beam relative to the desired HGM distribution at the position of laser crystal. The transverse position of the end pumping beam is situated at one of the brightest spots of desired HGM to achieve an overlap between the desired HGM distribution and effective gain region at laser crystal. This approach sets the desired HGM to be a cavity mode with a higher gain than other cavity modes. However, the gain region overlaps among the specified HGM and also several other cavity modes, e.g., tilted HGMs, LGMs, or IGMs, etc. Moving the opaque-wire to be situated at the outer most nodal lines of the specified HGM selects the lasing modes from all gain-competing modes. With this cavity configuration, all other gain-competing modes suffer significant energy loss in one round-trip of the cavity due to diffraction by the opaque-wire, while the specified HGM suffers little diffraction energy loss. This approach allows all other gain-competing modes to have a much higher round-trip loss than the specified HGM. With this “mode-gain control” mechanism, any specified high-order HGM can be successfully excited in the resonator by controlling the inserted opaque-wire position (i.e., modal loss area) and the pump-beam position (i.e., modal gain area).
2.2 Simulation model of opaque-wire-inserted half-symmetric laser resonator
Here, let us perform simulation to verify if the proposed approach to excite a specified high-order HGM in end-pumped SSLs is practicable. In this study, we modified a simulation code of optical resonator, which was originally developed for the calculation of selective-excitation of the Ince-Gaussian modes in end-pumped solid-state lasers . The code was drafted by the software MATLAB  to simulate laser oscillation of an end-pumped solid-state system, The laser-oscillation simulation method this code adopts is based on Endo’s simulation method [24, 25], which can simulate a single-wavelength, single/multi-mode oscillation in stable/unstable laser cavities. A detailed description of the simulation method could be found in Ref. 5. The following description summarizes the simulation method used in this study. The method simulates the initial stimulated field with a partially coherent random field  in the space-frequency domain to avoid dependence between the initial field selection and the conversion field in a stable laser cavity. The initial stimulated field propagating back and forth in the resonator was stimulated by Fresnel-Kirchhoff integration . Besides, in this model, the effects of gain medium and optical elements (i.e., plane mirror and concave surface) are easily introduced by changing the optical field at each position . In summary, the code simulated the process of the actual lasing process from initial spontaneous random field. After a certain number of iterations, according to the boundary condition, the cavity will find the lasing mode distribution, which satisfies
As Fig. 1(a) shows, this study models the optical filed oscillation in an opaque-wire-inserted half-symmetric laser resonator. This cavity is formed by one planar mirror (i.e., coated face of laser crystal) and a concaved mirror with a curvature radius of R2 = 15 cm at a distance of L = 10 cm from the planar mirror. The planar mirror is provided by a high-reflection coated surface of a 1-mm thick laser crystal; this study assumes the refractive index of the crystal to be the index of Nd:GdVO4, n = 2. The lasing wavelength λ is selected to be the lasing wavelength of the Nd:GdVO4 laser, 1064 nm, which is commonly used in bio-sample manipulation. The opaque-wire of width around 20 μm is situated in front of the concave mirror in a distance of 2.5 cm.
2.3 Numerical results and discussion
Figure 2 , the resulting progresses of three different HGMs, HG2,2 mode, HG3,1 mode and HG4,5 mode, from opaque-wire-inserted resonator, show how the “mode-gain control” mechanism successfully selects a specified high-order HGM from an initial random field. All figures in Fig. 2 are output beams form the output coupler of the resonator. Refer to Fig. 2, the random field first converges to a source wave at the gain region due to the gain at one of the brightest spots of the specified HGM. The optical source at this region then creates an image source at the point symmetric to the optical axis of laser cavity. After that, all the oscillation optical fields seem to transfer to the interference pattern of these optical sources. The round-trip optical paths between these optical sources determine which transverse point of the transient pattern is bright or dark. Interestingly, we found that oscillation fields will oscillate toward to IGM-like patterns at first. Then, the oscillation filed will slowly end up with the desired HGM with gain-competing modes being dead through the introduction of significant diffraction energy loss by the opaque-wire during the oscillation process. Finally, the oscillation patterns converge to steady specified HGM output when the round-trip gain of the mode equals the round-trip loss. Figure 3 are movies show detail mode convergence of HG2,2 mode, HG3,1 mode and HG4,5 mode, respectively (one frame per two round trips, 4 frames per second).
Figure 4 shows several simulation resulting selective HGMs patterns from the proposed opaque-wire-inserted resonator. The resulting HGM qualities, mode purities, are also shown in the Fig. 4. The mode purity of the generated HG beam is defined as the degree to which the intensity pattern reproduces the theoretical HGM [29, 30]. The mode purity, or relative weight of the specified HG in the beam, is given by the convolution between the normalized resulting field u and the normalized HG field uHG, i.e.,
All the resulting HGMs from the proposed approach possess high mode purities greater than 97%.
3. Experimental results
3.1 Experimental setup and operation
Figure 5 depicts the experimental setup to the HGM generation from end-pumped SSLs, where the dimensions of the setup are based on numerical simulations. The experiments were carried out using an LD-pumped 1-mm-thick 3 at.% Nd-doped a-plate Nd:GdVO4 crystal (refractive index n = 2.0). One end-surface of Nd:GdVO4 crystal was coated to be transmissive at the laser-diode (LD) pump wavelength of 808 nm (>95% transmission) and highly reflective (99.8%) at the lasing wavelength of 1064 nm. A concave mirror R2 (radius of curvature: 15 cm) was coated to be 1% transmissive at 1064 nm. This mirror was placed 10 cm away from the laser crystal. The reflective side of laser crystal and the concave mirror formed a half-symmetric laser resonator. The laser crystal was attached to a Cu heat sink with a 3-mm-diameter hole. A 16 μm wide copper wire was mounted on a continuous rotation mount situated at a two-axis translation stage and placed approximately 2.5 cm away from the concave mirror. An elliptical LD beam was transformed into a circular one and focused onto the Nd:GdVO4 crystal using a microscope objective lens with a numerical aperture of 0.65. This produced a tight focus with a minimum spot size of approximately 30 μm at the crystal. The off-axis position and the shape of the pump beam at the laser crystal were controlled by slightly tilting the objective lens. The orientation and position of the metal-wire were controlled by the continuous rotation stage and two-axis stage, respectively. The absorption coefficient for the LD wavelength of 808 nm was 74 cm−1. The resulting absorption length was as short as 135 μm. In all cases, linearly π-polarized emissions along the tetragonal c-axis were observed. It should be noted that other laser crystals (such as Nd:YAG or Nd:YVO4) can also be used to generate high-order HGMs. We use Nd:GdVO4 in the experiment for the reason that Nd:GdVO4 is a compromise choice for high power output DPSS lasers. That is, Nd:GdVO4 crystals have higher slope efficiency than Nd:YAG crystals and provide better thermal conductivity and higher power output than Nd:YVO4 crystals.
The following description describes the experimental operations to the high-order HGMs excitation. First, shifting the end-side tight pumping beam focus laterally from the laser cavity optical axis produced a higher-order lasing IGM with high ellipticity. The lasing IGMs could be treated as a superposition of several HGMs with the same order . Thus, the lasing IGMs determined the orientation of the metal wire. The metal wire was then moved into the laser resonator and carefully adjusted by its transverse positions. The oscillation field converged to the HGM when the metal wire was situated at the nodal line position of a specified HGM. Once achieving a single high-order HGM lasing (e.g., HG6,4 mode), the proposed design can easily achieve other HGMs of the same nodal-line position (e.g., HG7,4 mode, HG8,4 mode and etc.) by slightly changing the pumping beam position along the direction of the metal wire.
3.2 Resulting high-order HGMs from end-pumped SSLs
Figure 6 presents the high-order lasing HGMs patterns produced by the experimental end-pumped SSLs. Controlling both the end-side pumping beam and metal-wire makes it possible to generate high-order HGMs lasing from end-pumped SSLs. The straight yellow lines in Fig. 6 represent the orientation and relative transverse position of the inserted metal-wire to the lasing high-order HGMs. The HGM patterns in Fig. 6(a) are the result of different metal wire orientations. The orientations of the nodal lines of a lasing HGM are parallel or orthogonal to the orientation of the inserted metal-wire. Note that an original “empty resonator”, i.e., the resonator that formed by a plane mirror (coated laser crystal) and a concave mirror without inserting the metal‐wire, contains rotational symmetry. That is, there is no limitation on the transverse mode pattern direction of any cavity mode of the empty resonator. The resonator rotation symmetry is broken by the inserted metal‐wire. The direction of the nodal lines of lasing high‐order HGMs, i.e., the x-y axis of lasing high-order HGMs, is defined by the inserted metal‐wire. The HGM patterns in Fig. 6(b) were produced by changing the end-side pumping beam focus position along the direction of the inserted metal wire while keeping the metal wire fixed (i.e., maintaining the same orientation and transverse position). Results show that once a lasing high-order HGM lasing was achieved, we can easily achieve another higher order HGM lasing by simply changing the off-axis pumping beam position.
Figure 6(c) shows some selected experimental lasing HGn,n mode patterns. The distribution of n × n bright spots of HGn,n modes is well-aligned in an array manner. This implies that the HGn,n modes have the potential to be applied in multiple-trap laser tweezers [14–16]. Figure 6(d) shows some selected experimental lasing HGn,n + 1 mode and HGn + 1,n mode patterns. Section 4 will show that this type of high-order HGMs can transform into square vortex array laser beams after passing through the Dove prism-embedded unbalanced Mach-Zehnder resonator .
Figure 6 shows that using “off-axis pumping” approach with a metal-wire-inserted resonator, one can successfully achieves high-order HGMs form an end-pumped solid-state laser. Note that without inserting an opaque-wire into the resonator, multi-modes beams or other kinds of general cavity modes, such as Ince-Gaussian modes , Laguerre-Gaussian modes  and the two specific types of high-order HGMs: HG0,n modes and HGn,0 modes , will be excited from end-pumped solid-state lasers with off-axis LD pumping.
Figure 7 shows the laser performances: output/input power characteristics for each lasing high-order HGMs shown in the Fig. 6. Generally, a high-order HGM have an output/input power of a smaller value due to it suffers a higher round-trip diffraction loss than a low-order HGM. Besides, the output/input power of a lasing HGM is also related to the experimental operation. That is, the output/input power will be influenced by the overlap degree between the focusing pumping beam at laser crystal and the brightest spots of a specified lasing HGM. Indeed, changing pumping beam power resulted in the change of the effective gain region at the laser crystal, leading to the change in the lasing status of a lasing high-order HGM. Consequently, we didn’t measure the slope efficiency of all lasing high-order HGMs but only show single output/input power of lasing high-order HGMs.
For practical application concern, we would like to know how the inserted metal-wire influences the output lasing HGMs’ beam power. Concerning achieving high-order HGM-lasing from end-pumped SSLs, only two kind of high-order HGMs (HG0,n and HGn,0 modes) can be directly generated from end-pumped SSLs from a laser resonator without metal-wire inserted [7–10]. Thus, this study measured and compared output/input power of lasing HGn,0 modes from the same resonator without/with the metal-wire inserted. Figure 8 shows the output/input power of all lasing higher-order HGn,0 modes (n = 1 to 4). The output lasing HGMs’ beam power from the same resonator without/with the metal wire inserted are almost same. The results shows that situating a tiny metal-wire at the nodal-line positions of a lasing HGM will not cause much diffraction loss to the oscillated HGM, i.e., situating a tiny metal-wire at the nodal-line positions of a lasing HGM will not lead to significance influence in laser efficiency.
4. Converting HGn,n + 1 mode or HGn + 1,n mode to square vortex array laser beams by the Dove prism-embedded unbalanced Mach-Zehnder interferometer 
4.1 Create square vortex array laser beams
Superposing a special HGM pair, HGn,n + 1 mode and HGn + 1,n mode, can result in a square vortex array laser beams by controlling a relative phase among the pair of HGM modes. Figure 9(a) shows the relationship between the two HGMs and the resulting square vortex array laser beam. While a HGM pair, HGn,n + 1 mode or HGn + 1,n mode, are superposing with ± π/2 phase difference, the resultant superposed beam will be a vortex array laser beam which contains n2 + (n + 1)2 vortices that well aligned in a square manner. The effect of the π/2 phase difference between the two sub-beams introduces a coefficient i between two superposed sub-beams. Thus, the resulting square vortex array laser beam UVL is
Passing a HGn,n + 1 mode or a HGn + 1,n mode through the Dove prism-embedded unbalanced Mach-Zehnder interferometer we proposed recently  can convert a HGn,n + 1 mode or HGn + 1,n mode into square vortex array laser beams. Figure 9(b) illustrates the schematic diagram of square vortex array laser beam generation. The detail description of the structure of the interferometer could be found in Ref. 17. The embedded Dove prism rotates about the optical axis, the z-axis, at 45 degrees. Passing a nearly-collimated HGn,n + 1 laser beam through the interferometer produces a vortex array laser beam with n2 + (n + 1)2 vortices embedded in it. This vortex array is generated by precisely controlling the relative π/2 phase difference between the HGn,n + 1 mode and its rotated mode, namely the HGn + 1,n mode; i.e., controlling the position of the reflection mirror (e.g., a commercial nano-stage with a precision motorized actuator with < 100-nm resolution).
The operation of the interferometer is detailed as follows. An incident HGn,n + 1 laser beam splits into two sub-beams after passing through the beam splitter (BS1). One sub-beam arrives at the CCD with maintained incident HGn,n + 1 mode orientation, which is reflected by two mirrors and the second BS. The other sub-beam arrives at the CCD with HGn,n + 1 mode distribution with 90 degrees rotation, i.e., the same field distribution as the HGn + 1,n mode distribution. The rotation effect is created by passing the sub-beam through the rotated Dove prism. The Dove prism has a very interesting effect on the orientation of incident beams. If the Dove prism rotates to angle θ, the beam passing through the Dove prism rotates to angle 2θ . Thus, after passing through a Dove prism rotated at 45 degrees, the sub-beam rotates about the optical axis at 90 degrees. The two returned sub-beams are recombined into one beam after passing through the second beam splitter (BS2). To prevent any mismatch between the wave-front curvatures of both modes combined at the CCD, the optical lengths along both paths are nearly the same, with only a π/2 phase difference between the two sub-beams. This could be achieved by precisely controlling the mirror position with a commercially available nano-stage. Since the HGn,n + 1 modes and the HGn + 1,n modes differ only in a rotation of 90 degrees, passing HGn + 1,n modes through the interferometer will result in a square vortex array laser beam.
Figure 10 shows two resulting vortex array laser beam distributions from the incident HGn,n + 1 modes or HGn + 1,n modes with different mode numbers n = 1 and n = 2. Figure 10(a) and 10 (b) show the intensity distribution of selective oscillation HGn,n + 1 modes or HGn + 1,n modes. Figure 10(c) and 10(d) show the resulting square vortex array laser beams produced by the interferometer. The symbol z in the Fig. 9 denotes the distance from the CCD to the output port of the interferometer, BS2. The experimental results show that the resulting vortex array laser beam can maintain its vortex array pattern while it is propagating. The number of optical vortices embedded in each square vortex array laser beams are 5 and 13, respectively.
4.2 Discussions on square vortex array laser beam properties
The embedded vortex number of the resulting vortex array laser beam can easily be calculated by counting the number of dark points in the resulting field distribution UVL. The well-known normalized HGMs distributions are
Since Hermite polynomials have only real values, the Eq. (5) thus becomes
Note that the solution of Eq. (6) is symmetric about x-y = 0, and Hn(x) and Hn + 1(x) cannot be zero with the same x value. Therefore, the solutions of the equation UVL = 0 are the solutions of
The solutions of Eq. (7) are the n2 cross-section points of x-nodal line of HGn,n + 1 mode and y-nodal line of HGn + 1,n mode, and the solutions of Eq. (8) are the (n + 1)2 cross-section points of y-nodal line of HGn,n + 1 mode and x-nodal line of HGn + 1,n mode. Thus, the total number of vortices in the resulting vortex array laser beam is n2 + (n + 1)2.
Figure 11(a) shows analytical vortex array laser beam produced by superposing an analytical HG10,11 modes and HG11,10 modes with π/2 phase difference. Figures 11(b) and 11(c) show the corresponding enlarged amplitude and phase distribution of the square vortex array laser beam center patterns. To verify that the dark spots in the resulting laser beam are actually vortices, this study further simulates an interferogram resulting in vortex array laser beams with a tilted plane wave (Fig. 11(d)), which further indicates the cross-section positions and the order of vortices in the resulting vortex array laser beams. In Fig. 11(d), the characteristic forks in the interferogram are all one fringe split into two, i.e., these vortices are all first order vortices. The red and blue spots in Fig. 11(c) and 11(d) indicate the positions of vortices, which are the dark spots appearing in Fig. 11(b).
The two portions of the vortices (i.e., solutions of Eq. (7) and Eq. (8)) are aligned like a checkerboard, as depicted by the blue and red spots in Fig. 11(c) and 11(d), respectively. In other terms, the direction of optical torque is reversed for adjacent vortices. This well-aligned vortex array can be used to construct a new class of multiple-trap optical tweezers and atom traps in the form of two-dimensional arrays. It should be noted that there is no restriction on the number of embedded vortices in the resulting square vortex array laser beam. Simply increasing the mode number of the incident HGn,n + 1 modes or HGn + 1,n modes can increase the resulting vortices number in the resulting square vortex array laser beams. Besides, it is promised that the square vortex array laser beams are potential to the usage in laser tweezers and other applications, such as atom guiding, since such vortex array laser beam repeats its vortex array pattern while focusing. It is because the well-known property of Gaussian beams tells us: the Fourier transform of a Gaussian near-field pattern, i.e., the far-field pattern, exhibits a Gaussian distribution and any Gaussian beam can repeat itself while focusing. In this case, the focused pattern of the square vortex array laser beam is simply a coherent superposition of two focused Gaussian beams, the HGn,n + 1 mode and its rotated replica HGn + 1,n mode with a constant π/2 phase shift, thus resulting in the same square vortex array pattern.
This study presents the first systematic approach to exciting all high-order Hermite-Gaussian modes (HGMs) in end-pumped solid-state lasers with an opaque-wire-inserted laser resonator. The excitation of high-order Hermite-Gaussian modes is based on the “mode-gain control” concept. This study presents numerical simulations and experimental results for exciting HGMs in end-pumped solid-state lasers. It is noteworthy that the bright spots of one type of high-order HGMs, the HGn,n modes, are well-aligned in an array, showing great potential for multiple-trap optical tweezers [14–16]. This study also demonstrates an approach to create square vortex array laser beams by passing HGn,n + 1 or HGn + 1,n modes through the Dove prism-embedded unbalanced Mach-Zehnder interferometer . The resulting square vortex array laser beams contain n2 + (n + 1)2 vortexes aligned in a square manner, and can maintain a vortex array pattern during focusing. The square vortex array laser beams possess such properties show potential to dark multiple-trap optical tweezers and atom traps in the form of two-dimensional arrays, or to study the transfer of angular momentum to micro particles or atoms (Bose-Einstein condensate).
This work was supported by the Advanced Optoelectronic Technology Center, National Cheng Kung University, under projects from the Ministry of Education and the National Science Council (NSC 99-2112-M-006-007-MY3) of Taiwan.
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