This study reports a possible first systematic approach to the selective excitations of all Mathieu-Gauss modes (MGMs) in end-pumped solid-state lasers with a new kind of axicon-based stable laser resonator. The study classifies MGMs into two categories, and explores and verifies the approach to excite each MGM category using numerical simulations. Controlling both the “cavity mode gain” and the “cavity conical asymmetry” of the axicon-based stable laser resonator achieves the proposed selective MGM-excitation approach.
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Nondiffraction beams have attracted considerable attention since Durnin et al. first discovered the generation of Bessel beams in 1987 [1–2]. In addition to Bessel beams, Mathieu Beams are another family of non-diffraction beams that are solutions of paraxial wave equation in the elliptic coordinate , used in localized X-waves , photonic lattices , and for transferring angular momentum . Because the properties of ideal non-diffraction beams possess an infinite energy distribution that is physically un-realizable, researchers have endeavored to describe nearly non-diffraction beams that can propagate a long distance without significant divergence, particularly Bessel-Gauss (BG) beams, which were Bessel beams apodized by Gaussian transmittance . Bessel-Gauss beams carry finite power that can be realized experimentally to a very good approximation. Similar to BG beams, Mathieu-Gauss modes (MGMs) are another kind of nearly non-diffraction beams, which were the ideal non-diffraction Mathieu beams apodized by Gaussian transmittance, which carry a finite power that can be realized experimentally to a very good approximation . The nearly non-diffraction properties of MGMs shows their potential to practical applications in optical interconnections , laser machining [10,11], collimation and measurement , optical manipulation [13,14], and etc.
In 2008, Alvarez-Elizondo et al. first observed MG modes (MGMs) directly generated in an axicon-based stable resonator in a real CO2 laser by slightly breaking the symmetry of the cavity . Using special micro-grain Nd: YAG laser crystals, Tokunaga et al. also observed spontaneous MGMs oscillation in end-pumped solid-state lasers owing to the special fluorescence anisotropy effect of micro-grain laser crystals . However, a general approach for exciting any specified MGM in a laser system has yet to be discovered. This study investigated finding a way to excite a specified MGM in an end-pumped solid-state laser system. We drafted codes to simulate the lasing operation of an end-pumped solid-state laser system to explore selectively exciting a specified MGM in end-pumped solid-state lasers using numerical simulation. This paper proposes an axicon-based stable laser resonator, and reports what might be the first systematic method of generating specified MGMs in an end-pumped solid-state laser system. Controlling “cavity mode gain” and “cavity conical asymmetry” of the axicon-based laser resonator achieves selective excitation of MGMs.
Section 2 briefly describes MGMs for getting physical characteristics of MGMs. Section 3 details the simulation model of the axicon-based stable laser resonator. Section 4 provides numerical exploration of the selective excitation of MGMs. Section 5 shows the resulting lasing MGM properties. Section 6 presents further discussions of this study. Section 7 gives a brief conclusion of this study.
2. Basic formalism of Mathieu-Gauss Modes
Mathieu-Gauss modes are Mathieu beams apodized by Gaussian transmittance, and are written as the product of three factors: a complex amplitude depending on the z coordinate only, a Gaussian beam, and a complex version of the transverse shape of the non-diffraction Mathieu beam . The complex amplitude of the m-th order even and odd MGMs propagating along the positive z direction of an elliptic coordinate system r = (ξ, η, z) are described in the Mathieu function by 
The elliptic coordinate (ξ, η) is related to the Cartesian coordinate (x, y) as x+iy=f0cosh(ξ+iη), where ξ∈ [0, ∞], η∈ [0,2π], and f0 is the semifocal separation at the Gaussian beam waist plane z=0. The and are the m-th order even and odd radial Mathieu functions, and and are the m-th order even and odd angular Mathieu functions. is the fundamental Gaussian beam,, w0 is the Gaussian beam width at the waist plane, k is the wave number, and kt is the transverse wave number. The MGMs are characterized by the ellipticity parameter . When, the foci of the elliptic coordinates collapse at the origin, and the nodal line of MGMs will approach to circular nodal lines. The main difference between even and odd MGMs is that odd MGM patterns break along the x-axis, but even MGMs patterns do not.
Mathieu Gauss modes form as a superposition of fundamental Gaussian beams, whose mean propagation axis lie on the surface of a cone, whose amplitudes modulate by angular Mathieu functions . For this reason, MGMs possess two special characteristics. First, MGMs retain non-diffracting propagation properties from the non-diffraction Mathieu beam within the range z ∈ [- zmax, zmax] while the beam parameter γ>>1. The definition of the parameter γ is , where θ 0 is the half-aperture angle of the cone surface and θ G is the diffraction angle of constituent Gaussian beams, and . Figure 1 (a) shows the transverse amplitude distribution of a m=2, q=5 MGM at positions: z=0, z=0.6zmax and z=1.2zmax. Figure 1 (b) shows the propagation amplitude patterns along the planes (y, z) and (x, z) in the range [-1.6 zmax, 1.6 zmax], which demonstrate the non-diffraction properties of MGMs. The second characteristic of MGMs is that they retain their transverse shape of the power spectrum to a circular ring during beam propagation. The mean radius and width of the annular ring are determined by kt and w 0 respectively . Figure 1(c) shows a general circular shape spectrum of an analytical m=2, q=5 even MGM.
3. Simulation of Mathieu-Gauss Mode oscillation in an axicon-based stable laser resonator
This study drafted codes using the software MATLAB  to simulate laser mode oscillation of an end-pumped solid-state laser system with the proposed axicon-based stable laser resonator, and explored selectively exciting a specified MGM in an end-pumped solid-state laser system using numerical simulation.
3.1 Axicon-based stable laser resonators
Figure 2 is the scheme diagram of the proposed axicon-based stable laser resonator for selective MGM excitation in end-pumped solid-state lasers. For usage in end-pumped solid-state laser systems, the cavity forms by a convex-concave lens and a bi-conical lens. Two surfaces of convex-concave lens, R1 and R2, are set in the same radius curvatures R for avoiding defocus of the end-side pumping beam. The bi-conical lens form by two conical surfaces, C1 and C2, set as different tilted angles α and β. The right concave surface of the convex-concave lens R2 and the left surface of the bi-conical lens C1 are coated reflectively, constituting a general axicon-based stable laser resonator . Within the cavity, each plane-wave component of the MG beam will see a finite Fabry-Perot resonator, and the radius of curvature of the concave mirror R is a free parameter useful in changing the diffraction characteristics of output lasing MG beams . Reference 18 presents a detailed illustration of the general axicon-based stable laser resonator.
General usage of an axicon-based stable laser resonator involves choosing the concave surface as the output coupler of the resonator . In such a situation, the emitting laser beams retain non-diffraction properties in a distance around zmax. Different from general usage, this study chose the left conical surface of the bi-conical lens, C1, as the resonator output coupler. In MGM excitation, the end-side pumping beam focuses on the crystal in the resonator close to the convex-concave lens. While achieving MGM oscillation in the laser resonator, the resulting optical field distribution at the conical surface C1 in the resonator is close to the spectrum of the lasing MGM. The tilt angle of the right conical surface C2, β, is close to twice the tilt angle of the left conical surface C1, α, to provide the same optical phase change as the reflective conical surface C1 provides. That is, the oscillation field changed by transmitting the right conical surface C2 is just like the oscillation field being reflected by left conical surface C1. In the proposed pumping scheme, emitting MGMs from the laser resonator will retain non-diffraction properties at a distance up to 2 × zmax.
3.2 Description of the simulation approach
This study performed simulation to explore how to excite a specified MGM in an end-pumped solid-state laser system. In this study, we modified a simulation code of optical resonator, which was originally developed to 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 [20,21], which can simulate a single-wavelength, single/multi-mode oscillation in unstable/stable laser cavities. A detailed description of the simulation method could be found in Ref. 19. 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., concave surface and conical 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 stimulated 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. 2 shows, the study models laser oscillation in an axicon-based stable laser resonator, formed by two reflective surfaces of the convex-concave lens and the bi-conical lens. In the simulation, the curvature radius of the concave surface R2 is set with the value R=2m and the tilting angle of the two conical surfaces is set according to the selection-modes. The reflective conical surface is set at a distance of L = 25 cm from the reflective concaved mirror. This study assumed the refractive index of the crystal to be the index of Nd:GdVO4, n=2, and set the peak emission wavelength of the laser to be 1064 nm. In simulation, the laser crystal was assumed to directly contact with the concave mirror.
4. Controlled Mathieu-Gauss mode excitation
This study classified all MGMs into two categories, according to MGM characteristics, including (1) q=0 MGMs and (2) q>0 MGMs. Figures 3 (a) and (b) show some typical transverse patterns of two categories of MGMs. The q=0 MGMs have circular nodal lines, while q>0 MGMs have elliptical nodal lines. The following subsections detail the scheme to excite each category of MGMs.
4.1 q=0 MGMs excitation
Using the axicon-based stable laser resonator, we can only confirm that the resulting lasing field from the laser resonator will be a conical wave superposed field, apodized by Gaussian transmittance. However, both Bessel-Gauss modes (BGMs) and MGMs can survive in this cavity. For selective excitation of a specified order of single q=0 MGM oscillation in the resonator, we provide a condition in which the specified order q=0 MGM in the end-pumped solid-state laser system will have a much higher round-trip gain than other MGMs and BGMs. (It should be noted that the BGMs we mention here is based the definition in Ref. 7 and Ref. 8. The patterns of the BGM transverse cross-sections are ring shape.) Under such a condition, after several round-trip oscillations, only the specified order q=0 MGM will survive in the resonator. To create this lasing condition, we checked the analytical field pattern of q=0 MGMs and then caused the effective gain region (i.e. focusing pumping beam transverse position) to overlap with one of the target spots of the specified q=0 MGM distribution at the position of the laser crystal in simulation. Referring to Fig. 4 , the target spot of a q = 0 MGM was chosen as one of the inner/brightest spots of the specified order q=0 MGM. In simulations, the gain regions were chosen as the circular shape estimated by the analytical field pattern of q=0 MGMs. In real experiments, tuning the end-side pumping beam can easily control the effective gain region at the laser crystal.
This study used simulations to verify if this “gain region control” mechanism to excite a specified q=0 MGMs is practicable. The tilt angle of the reflective surface C1, α, was set as 0.2 degree. We first solved the target point transverse position of MGMs at the laser crystal position from order m=1 to m=3 using Eq. (1) and (2). Then, we plotted MGMs patterns to estimate the gain region size. Table 1 shows the parameters of the estimated gain region to MGMs from order m=1 to m=3. Figure 5 shows the resulting oscillation optical pattern from simulations with effective gain regions addressed in Table 1. The corresponding gain regions used in the simulations are plotted in Fig. 5 by red circles. The square window sizes of all figures in Fig. 5 are in dimensions of 3w0×3w0. Simulation results show that using the proposed axicon-based stable laser resonator accompanying the control of the end-side pumping beam position, can successfully excite a specified order q=0 MGM in end-pumped solid-state lasers. The simulation process revealed that while the end-side pumping beam overlaps the gain regions of two q=0 MGMs of adjacent order, two q=0 MGMs compete with each other for optical gain. The phenomenon becomes more common when the specified q=0 MGM is of higher order (the target spots of two adjacent high-order q=0 MGMs are closer). Using a pumping beam of a smaller beam size/gain region in the simulation can avoid such mode-gain competition phenomenon. From the simulation results, we can speculate that exciting a specified q=0 MGMs of high order would be easier in a real experiment, using an end-side pumping beam with a smaller focusing beam.
4.2 q>0 MGMs excitation
For exciting q>0 MGMs in the end-pumped solid-state laser with an axicon-based stable laser resonator, we first try the “gain region control” mechanism, i.e., we cause the end-side pumping beam/gain region to overlap with one of the target spots of the specified q>0 MGM distribution at the laser crystal position in simulations. Referring to Fig. 6 , the target spot of a specified q>0 MGM is one of the brightest spots of its field distribution. However, no matter how we change the effective gain region in simulations, we cannot achieve any q>0 MGM oscillation in an axicon-based stable laser resonator with axial symmetry. The simulation results suggest that to excite a specified q>0 MGM in an end-pumped solid-state laser system requires finding other differences between q=0 MGMs and q>0 MGMs, other than the target spot position.
Comparing q>0 MGMs with q=0 MGMs, the q>0 MGMs field distribution has non-circular asymmetry, i.e., the q>0 MGM filed distribution along the x-axis is different from the q>0 MGM field distribution along the y-axis. To excite q>0 MGMs, we introduce asymmetry into the axicon-based stable laser resonator. The approach to break the symmetric cavity involves replacing the bi-conical lens with an asymmetric bi-conical lens. The surface sag of the left reflectively asymmetric conical surface C1, zr, is described by
The slope of the asymmetric conical surface profile on the y-z plane is ρ, and the slope of the asymmetric conical surface profile on the x-z plane is σ×ρ. The asymmetry of the reflective conical surface is controlled by the cavity asymmetric parameter σ. The surface sag of the right transmit asymmetric conical surface C2, zt, is set for providing the same optical phase change as that the reflective asymmetric conical surface provided. The corresponding surface sag of the right transmit conical surface isFigure 7 (a) shows the cross-section of the bi-conical lens on the y-z plane, and Fig. 7 (b) plots the profiles of the reflective asymmetric conical surface C1 on the plane (y, z) and plane (x, z).
Using the asymmetric axicon-based stable laser resonators accompanying the “gain region control” mechanism, we can easily achieve a specified q>0 MGM excitation in end-pumped solid-state lasers. For selective excitation of a specified order of single q>0 MGM oscillation in the resonator, we cause the effective gain region to overlap with one of the target spots of the specified q>0 MGM distribution at the laser crystal position. Referring to Fig. 6, the target spot of a q>0 MGM was chosen as one of the brightest spots of the specified order q>0 MGM. With usage of the asymmetrical conical reflective surface, only the asymmetrical optical field can survive in the cavity (i.e., q>0 MGM). Accompanying the “gain region control” mechanism provides a condition where the specified order q>0 MGM in the end-pumped solid-state laser system will have a much higher round-trip gain than other q>0 MGMs. With such a condition, after several round-trip oscillations, only the specified order q>0 MGM will sustain in the resonator. To create this lasing condition, in simulations, the gain regions were chosen as an elliptical shape, estimated by the analytical field pattern of q>0 MGMs. In real experiments, the effective gain region at the laser crystal was controlled by tuning the end-side pumping beam. The azimuthally pumping scheme easily achieves an elliptical pumping beam/gain region .
This study used simulations to verify if using the “asymmetric axicon-based stable laser cavity” with “gain region control” mechanism to excite q>0 MGMs is practicable. In simulation, the tilt angle of the asymmetric conical surface profile on the y-z plane was set as 0.2 degree, i.e., the slope of the surface profile on the y-z plane, ρ, was tan(0.2°), and the cavity asymmetric parameter was set as σ = 0.98. We solved the target point transverse position of MGMs from order m=1 to m=3 at the laser crystal position using Eq. (1) and (2), followed by plotting MGM patterns to estimate the gain region sizes of each q>0 MGM. Table 2 shows the estimated gain regions of q>0 MGMs from order m=1 to m=3. Figure 8 shows the resulting oscillation optical pattern from simulations with effective gain regions addressed in Table 2. The corresponding gain regions used in the simulations are indicated in Fig. 8 by red ellipses.
Figure 9 are sampled movies which show detail MGM convergence of the two categories. The movie shows the time-variation amplitude distribution at laser crystal (one frame per ten round trips, 2 frames per second). The movies demonstrate how the “gain region control” mechanism successfully selects a specified MGM of the two categories from an initial random field pattern. The random field first converges to a source wave at the gain region due to the gain at the target region. The optical source at the target region then creates other image source at 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. Finally, the oscillation patterns converge to steady specified MGM output when the round-trip gain of the mode equals the round-trip loss of the mode.
5. Verification of lasing MGMs properties
MGMs have two specific properties; (1) they retain the non-diffracting propagation properties of the ideal non-diffraction Mathieu beam within the z range, and (2) the power spectrum for MGMs is given by , which corresponds to an annular ring . Beam-diffraction properties and the lasing beam spectrum from the proposed selective MGM excitation scheme are checked by simulations. Figure 10 shows propagation of the amplitude profile along plane (y, z) or plane (x, z) of several lasing MGMs from the laser output coupler, where the symbol d denotes the propagation distance from the conical surface C2. The lasing beam from the proposed approach retains non-diffracting propagation properties within a range of 2zmax. Figure 11 shows the spectrum of lasing MGMs from the proposed selective MGM excitation scheme and the analytical MGM spectrum for comparison. The spectrums of the resulting lasing modes from the proposed MGM-excitation approach (Fig. 11 (b)) are an annular ring shape, similar to the spectrum patterns of analytical MGMs (Fig. 11 (c)).
Note that changing the asymmetric conical surface slope ρ will not influence the resulting MGM q value, but will only influence the non-diffraction properties of MGMs, such as transverse wave number kt and MGM non-diffraction region 2zmax. The q value of the lasing q>0 MGMs from the asymmetric axicon-based laser resonator are related to the bi-conical lens asymmetry (i.e., cavity asymmetry parameters σ). Simulation results show that all resulting q>0 MGMs from a same asymmetric axicon-based stable laser resonator have a same q value. For example, all q>0 MGMs lasing patterns shown in the Fig. 8 to Fig. 11 are resulting from a same σ=0.98 asymmetric axicon-based stable laser resonator. Their q values are all 4.5. Figure 12 shows several resulting lasing m=2, q>0 MGMs from an asymmetric axicon-based stable laser resonator of different cavity asymmetry parameters σ. The results show the possibility of achieving a lasing q>0 MGM of higher ellipticity parameter q using an asymmetric axicon-based laser resonator with a smaller asymmetry parameter σ. Besides, we would like to note that the q value of MGMs is a continuous value. Even introduce small asymmetry into the laser resonator, the resulting lasing MGMs from resonator will become q>0 MGMs. Though we can always numerically distinguish the difference between a q=0 MGM and a q>0 MGM of small q value, it is very hard to visually observe the pattern difference between a q=0 MGM and a q>0 MGM of small q value (e.g. a q=0.2 MGM). Further simulation results show that only while cavity asymmetry parameters σ<0.998, we can visually observe that the resulting MGMs from the resonator are q>0 MGMs.
The proposed scheme is unlimited in its ability to excite higher order MGMs. However, simulating the lasing process of higher order MGMs requires more calculation memory and calculation time to describe the larger and finer field patterns. Figure 13 shows some simulated lasing patterns of higher order MGMs from the proposed selective MGM excitation scheme and its analytic patterns for comparison.
6. Further discussion
Note that replacing the convex-concave lens of the laser resonator with the planar mirror also achieves selective MGM-excitations in simulations. Refer to Fig. 2; while replacing the convex-concave lens with planar mirror, the non-diffraction region of the resulting MGMs will be a region that started from conical surface C2 of the bi-conical lens to the right-end of the rhombus. However, because such an axicon-based laser resonator with a planar reflective surface will introduce more diffraction loss , this study only discussed the laser resonator using concave-type reflective surface. Besides, using asymmetric bi-conical lens is not the only approach to introduce cavity asymmetry. Further simulation results found that inserting a cylinder lens of weak optical power into the proposed symmetric axicon-based stable laser cavity accompanying the “mode gain control” mechanism also achieves selective MGM excitation. However, because such a three-element configuration (convex-concave lens, cylinder lens and bi-conical lens) might be difficult to align and operate in a rear experiment, this study only discussed selective MGM excitation using asymmetric bi-conical lens.
For considering technical implications for experimental development, the tolerance to the MGMs selective-excitation scheme is considered here. From simulation explorations, we found that except to using the “gain region control” mechanism, another key point to the selective-excitation of a specified MGM is avoiding the gain region overlapping the target spot of the MGMs of neighbor order m. The tolerance to the pumping beam position and beam size is actually depending on the distance between two MGM target spots of neighbor order. For example, Fig. 14 plots analytical normalized intensity profiles along x-axis at the laser crystal of several q=4.5 even MGMs (order m=1 to m=8). All MGMs intensities shown in Fig. 14 are normalized by their target spot intensity. The figure gives a good illustration for the estimation of pumping beam position and pumping beam size. Besides, the figure can also be use to estimate the tolerance to pumping beam position and pumping beam size. Referring to Fig. 14, the distance between and is about 0.15w0, while the distance between and is about 0.08w0. When we use a pumping beam with focus diameter of 0.15w0, the tolerance of the pumping beam transverse position to excite the is about 0.07w0 while the tolerance of the pumping beam transverse position to excite the is about 0.01w0. The w0 value is depending on design of the laser resonator. Figure 14 also shows that to excite a high-order MGMs, we have to use a pumping beam of smaller beam focus, and the tolerance of the pumping beam position to excite a high-order MGMs will be smaller than the tolerance to excite a low-order MGMs. The tolerance to excite other MGMs can also be estimated in a similar way.
Besides, we note that some literatures refer to that when q→0, the elliptical MGMs reduce to circular BGMs . We include here explicit expressions of the BGMs in order to provide appropriate clarification. The ring-shape BGMs we mentioned in section 4.1 is following the BGMs definition in Ref. 7 and Ref. 8, i.e.,
When q→0, the elliptical and will reduce to and respectively. With the definition of even BGMs and odd BGMs, the q=0 MGMs excitation methods proposed in this paper is actually the method to the selective-excitation of even BGMs and odd BGMs.
This study proposed a systematic way to selectively excite Mathieu-Gauss modes in end-pumped solid-state lasers with a new configuration of an axicon-based stable laser resonator. This study classified MGMs into two categories: q=0 MGMs and q>0 MGMs, by the MGM ellipticity parameter q. Using the “gain region control” mechanism in the symmetric axicon-based stable laser resonator and the asymmetric axicon-based stable laser resonator easily achieves selective excitation of q=0 MGMs and q>0 MGMs. The ellipticity parameter q of a lasing MGM from the proposed resonator relates to cavity asymmetry. Numerical simulation verifies the non-diffraction property and the special characteristic of the annular spectrum of the lasing beam from the proposed selective MGM excitation schemes. The proposed resonator configuration and pumping scheme can achieve a lasing MGM of a specified mode order that maintains non-diffraction propagation distance twice as long as the older pumping scheme.
This work was supported in part by a grant from the National Science Council of Taiwan, R.O.C., under contract no. NSC 99-2112-M-006 −007 -MY3
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