This study report a first systematic approach to the selective excitations of all Ince-Gaussian modes (IGMs) in end-pumped solid-state lasers. The proposed Ince-Gaussian mode excitation mechanism is based on the “mode-gain control” concept. This study classifies IGMs into three categories, explores and verifies approach to excite each IGM category using numerical simulation.
©2007 Optical Society of America
Two transverse lasing modes, Hermite-Gaussian modes (HGMs) and Laguerre-Gaussian modes (LGMs) have been widely investigated with analytical, numerical, and experimental techniques. These two modes separately form two complete families of exact and orthogonal solutions of the paraxial wave equation (PWE) in rectangular and cylindrical coordinates. Researchers have recently proposed a third complete family of transverse modes, PWE solutions in elliptic cylindrical coordinates [1, 2], namely the Ince-Gaussian modes (IGMs). In addition, Ince-Gaussian modes constitute the continuous transition modes between HGMs and LGMs [1, 2]. Many studies have recently explored the characteristics of this novel IGMs [3–6]. Formulating approaches to produce predicted IGMs in real laser systems is an important step in exploring IGM properties and its potential applications. Researchers have recently observed IGMs in a stable resonator in a LD-pumped solid-state laser by breaking the symmetry of the cavity under tight pump beam focusing conditions [7–9]. These attempts include introducing a cross hair inside the cavity  or adjusting the azimuthal symmetry of the short laser resonator [8–9].
A general rule for exciting any specified IGM in a laser system has yet to be discovered. This study drafts code to simulate the lasing operation of an end-pumped solid-state laser system, and then studies the resulting mechanism of IGM excitation in the end-pumped solid-state laser using numerical simulation. This paper reports what may be the first systematic method of generating all IGMs in a laser system. This study proposes a selective excitation of desired lasing transverse modes for all IGMs, exhibiting pair(s) of the “target” lasing spots, with a pump beam whose profile matches one target spot.
2. Basic formalism of Ince-Gaussian modes
The Ince-Gaussian modes propagating along the z axis of an elliptic coordinate system r=(ξ,η, z), with mode number p and m and ellipticity ε, are given by 
where the elliptic coordinate is defined in a transverse z plane as x=f(z) cosξ cosη, y=f(z) sin ξ sin η, and ξ∈[0, ∞], η∈[0, 2π]. f(z) is the semifocal separation of IGMs defined as the Gaussian beam width, i.e., f(z)=f 0 w(z)/w 0, where f 0 and w 0 are the semifocal separation and beam width at the z=0 plane, respectively. w(z)=w 0 (1+z 2/zR 2)1/2 describes the beam width, zR=kw 2 0/2 is the Rayleigh length, and k is the angular wave number of the beam. The terms C and S are normalization constants, the subscripts e and o refer to even and odd IGMs, respectively. Cmp(., ε) and Smp(., ε) are the even and odd Ince polynomials  of order p, degree m, and ellipticity parameter ε, respectively. In Eq. (1) and (2), r is the radial distance from the central axis of the cavity, R(z)=z+z 2 R/z is the radius of curvature of the phase front and Ψz(z)=arctan(z/z R). The parameters of ellipticity ε, waist w 0 and the semifocal seperation f 0 are not independent, but related by ε=2f 2 0/w 2 0. IGMs patterns can be recognized by 2 rules: degree m corresponds to the number of hyperbolic nodal lines and (p-m)/2 is the number of elliptic nodal lines. Figure 1 plots some analytical patterns of the IGp,m modes obtained from Eq. (1) and (2).
3. Simulation model of a half-symmetric laser resonator
This paper uses codes that drafted by the software MATLAB  to simulate laser oscillation of an end-pumped solid-state system. The laser oscillation simulation model adopted in this study is based on Endo’s simulation method [11–13], which simulates a single-wavelength, single/multi-mode oscillation in unstable/stable laser cavities [11–13]. This code was recently used to demonstrate how a single IGe p,p mode pattern can be selected using azimuthal tight focus pumping . This study uses the code to explore how to excite a specified IGM in an end-pumped solid-state laser system [7–9].
As Fig. 2 shows, this study models a half-symmetric laser resonator similar to that in real experiments [7–9]. This cavity is formed by one planar mirror and a concaved mirror with a curvature radius of R 2=20 cm at a distance of L=10 cm from the planar mirror. The planar mirror is actually a high-reflection coated surface of a laser crystal; this study assumes the refractive index of the crystal to be the index of Nd:GdVO4, n=2. 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 stimulated initial field propagating back and forth in the resonator is mimicked by the Fresnel-Kirchhoff integration . In this model, the effects of gain medium and all optical elements could be easily introduced with an optical field change  at each position. The loaded gain at each station is assumed to be homogeneously broadened. The gain medium is simulated as several gain sheets and the saturated gain at each gain sheet i is expressed as
where gi(x,y) is the loaded gain, gi 0(x,y) is the small signal gain, and Is(x,y) is the saturation intensity. This paper assumes that Is≈1 kW/cm2, which is relevant to general solid-state lasers . The symbols Ĩ + i and Ĩ - i are the average right-going and left-going optical intensities defined as
Here, I + i(q) and I - i(q) denote the intensities of the qth iteration step, and α is a summation over a period of the cavity’s photon decay time. For example, this simulation sets the reflectivities of two mirrors at r1=100.0 % and r2=99%, respectively. Thus, the parameter α is given by r1r2=0.99.
The amplification of the optical field E(x,y) passing through gain sheet i with thickness d is expressed as
where the 1/2 is necessary because the gain is defined by the amplification of the optical intensity. With this simulation method, after a certain number of iterations, according to the boundary condition, the cavity will find the lasing mode distribution E(x, y) which satisfies
The simulation could find optical field distribution of the resulting lasing mode, thus is genuinely reflecting the phase structures of the modes as well as their intensity distributions. We have checked the validity of the simulation code by repeating simulations of a single mode oscillation in an unstable resonator and a multimode oscillation in a stable resonator in Ref. 11. The code was then used to study selective single IGM excitation in end-pumped solid-state laser systems.
4. Controlled IGMs excitation
This study classifies IGMs into three categories according to their characteristics, which lead to different resulting mechanisms. The three categories are: (1) IGe p,m modes (p≥m>0), (2)IGo p,m modes (p ≥m>1) and (3) IGe p,0 and IGo p,1 modes. Figures 3 (a), (b), and (c) show some typical patterns of the three categories of the IGMs, respectively. The square window sizes of all the figures in Fig. 3 are in dimensions of 8X8 times the spot size of the fundamental HG0,0 mode, w o. The IGMs of the first two categories have parabolic nodal lines, but Category 3 IGMs have only elliptical nodal lines. The first two categories share a same characteristic that the IGe,o m,m mode is actually the inner pattern of IGe,o p,m mode with parameter p>m. However, these IGMs are separated into two categories because only the IGo p,m mode patterns break along the X-axis-the IGe p,m modes do not. The following sections details schemes to excite each category of IGMs.
4.1 Exciting IGe p,m modes (p≥m>0)
Most IGMs observed in experiments [7–9] are IGMs of Category 1, IGe p,m modes (p≥m>0). To determine how to excite a specified IGe p,m mode (p≥m>0) in end-pumped solid-state laser system, this study checks its analytical pattern that are described in elliptic cylindrical coordinates [1, 2] first. As the red elliptic lines plotted in Fig. 4 indicate, the pattern focus is located at the brightest (i.e., target) spot of the IGe p,m mode. That is, when p=m, the focus is on the outermost spot of the pattern, and when p>m, the focus is on the outermost spot in the innermost elliptical nodal line. The focuses of these mode patterns are situated along the X-axis. These observations lead to the assumption that the mechanism to excite such IGe p,m modes with p≥m>0 in the experiments [7–9] is “to create a situation where the effective gain region overlaps with one of the target spots of the IGep,m mode distribution at the position of the laser crystal.”
This study uses simulations to verify if this “gain region control” mechanism to excite IGe p,m modes (p≥m>0) is practicable. First, solve the asymmetric transverse position of the most intense IGe p,m mode spots at the position of laser crystal of order p=5 and p=6 using Eq. (1). After that, plot the IGe p,m mode patterns to estimate the corresponding gain region, i.e., the region covering the most intense spot of IGM patterns. As the inset in Fig. 4 shows, the effective gain region was estimated by an elliptic region centered on the X-axis at a distance of x to the origin. Table 1 shows the parameters of the estimated gain region to the IGe p,m modes of order p=5 and p=6. Figure 5 shows the resulting oscillation optical pattern from simulations in all situations corresponding to Table 1.
The situation to excite IGe p,m modes (p≥m>0) proposed in this study was incidentally achieved in the experiments, either by shifting the output coupler sideways  or by tilting the cavity [8, 9]. Both approaches create an equivalent “off-axis pumping” mechanism letting the effective gain region be off-axis. In addition, controlling the pumping beam size/power and the crystal position along the longitudinal direction [8, 9] provide a mechanism to change the effective gain region size/shape.
4.2 Exciting IGop,m modes (p≥m>1)
This category of IGMs is rarely observed in real experiments [7–9]. To determine how to excite these cavity eigenmodes, the IGo p,m modes (p≥m>1), this study checks its analytical patterns first. As Fig. 6 shows, for IGo p,m mode with p=m, the most intense points of the pattern are in the two outermost spots of the pattern; while for IGo p,m mode with p>m, the most intense points of the pattern are in the two outermost spots in the innermost elliptical nodal line. As the inset in Fig. 6 shows, the “gain region control” mechanism was tried at first. However, simulations showed that if the relative gain region overlaps the most intense spot of the IGo p,m mode, the resulting IGM is still an IGe p,m mode of Category 1, but different in its relative X-axis direction of mode distribution. Results reveal that exciting IGo p,m modes with an elliptical-shaped pumping beam in a laser system using only the “gain region control” mechanism is not sufficient. Thus, to excite the IGMs of this category, another mechanism must be introduced.
To excite this category of IGMs, a condition must be created where “the losses of IGep,m modes are greater than IGop,m modes.” To create such a condition, the difference between the IGo p,m modes and the IGe p,m modes must be found. The pattern of IGo p,m modes differs from IGe p,m modes in breaking along the X-axis because their π phase change at that position. This study suggests putting a tiny opaque bar into the laser cavity situated on the optical axis along the X-axis of the crossed X-Y plane. With this cavity configuration, all IGe p,m modes will suffer significant energy loss in one round-trip of the cavity due to diffraction by the tiny opaque bar; however, IGo p,m modes will not. This means that using both the “gain region control” and “opaque bar insert” mechanisms together, any IGo p,m mode (p≥m>1) can be excited in an end-pumped solid-state laser system.
To verify if the suggested approach is workable, this study checks it by simulation. Similar to the approaches detailed in Section 4.1, solve the asymmetric transverse position of the most intense spots of IGo p,m with order p=6 and p=7 at the laser crystal using Eq. (1). Then plot the IGo p,m mode patterns to estimate the effective gain regions. As the inset of the Fig. 6 shows, the effective gain region is estimated by an elliptic region off-center from the origin of the crossed X-Y plane. Table 2 shows the parameters of the estimated relative gain region to the mode IGe p,m of order p=6 and p=7. As Fig. 1 shows, this simulation adds a relay station at the center of the laser cavity. An opaque bar with a width of 5% of the HG00 beam spot size was inserted at the relay station to block the passing oscillation optical field at the transverse position of the tiny opaque rod. Figure 7 shows the corresponding resulting oscillation optical patterns from the bar-inserted laser cavity with the controlled gain region addressed by Table 2.
IGMs of this category are formed very rarely and only in real experiments with a tilted cavity , i.e., azimuthal symmetry control. This agrees with our simulation experience, where the IGe p,m modes (p≥m>0) are more easily excited in an end-pumped solid-state laser system than IGo p,m modes (p≥m>1). Indeed, only three modes of this category, the IGo 4,2, IGo 5,5 and IGo 14,6 modes, are experimentally observed in anisotropic lasers, i.e., LNP and Nd:GdVO4 . This suggests that it may be possible to excite the IGo p,m mode (p≥m>1) without a opaque bar inserted into a laser cavity. In such a tilted cavity scheme , the IGo p,m mode (p≥m>1) is more efficiently excited through a highly asymmetric effective gain shape resulting from oblique pumping against the lasing axis. However, using the proposed approach, i.e., using both the “gain region control” and “opaque bar insert” mechanisms together, can excite any specified IGo p,m mode (p≥m>1) in an end-pumped solid-state laser system.
4.3 Exciting IGep,0 and IGop,1 modes
To determine how to excite specific IGe p,0 and IGo p,1 modes, this study checks their analytical patterns first. The IGe p,0 and IGo p,1 modes have a similar mode distribution. As Fig. 8 shows, the IGM patterns in this category have only elliptical nodal lines, but no parabolic nodal lines. In simulations, by controlling the elliptic gain region, which overlaps with spots of IGe p,0 and IGo p,1 modes, from center to outside one by one, only higher-order LGMs and IGe p,m modes are observed in the simulated oscillation patterns. Therefore, such an symmetric elliptical effective gain region cannot excite IGe p,0 and IGo p,1 modes in end-pumped solid-state systems. Checking the mode pattern Fig. 8 in detail reveals that: (1) most pattern spots are in half-elliptical form, (2) the outermost spots cover the largest area, thus occupying the largest mode volume. These two observations lead to the assumption that, “to excite IGep,0 and IGop,1 modes, the effective gain region at the laser crystal must be an asymmetrical shape that covers the outermost spots of the pattern.” This study assumes that the asymmetrical shape of the effective gain region in the simulation is the half-elliptical shape.
To verify if the suggested approach is workable, this study checks it by simulation. Solve the most intense point of the outermost spots of the IGe p,0 and IGo p,1 modes from order p=2 to p=7 at the laser crystal position using Eq. (1). Then plot the IGe p,0 and IGo p,1 mode patterns to estimate the size of the half-elliptical gain region. As the inset of the Fig. 8 indicates, the effective gain region is estimated by a half elliptic region that centered at the Y-axis at a distance of y to the origin. Table 3 shows the parameter of the estimated gain region of IGo p,m mode from order p=2 to p=7. Figure 9 shows the corresponding resulting oscillation optical patterns from the simulations with controlled gain region according to Table 3.
Note that the IGe p,0 and IGo p,1 modes are exceptional modes among IGMs. Their targeting spots are not the “brightest” spots of the pattern, since the brightest spot of such patterns vary according to its ellipticity parameter. The simulation in this study indicates that to excite the IGe p,0 and IGo p,1 modes, the shape of the target area should be selected asymmetrically that matches largest outermost spot most effectively. The asymmetrical gain region prevents excitations of undesired IGMs of Category 1. Note that some IGe p,0 and IGo p,1 mode patterns are frequently observed in Otsuka’s experiments with a tilted cavity [8–9]. In that experiment, the focusing light cone of the laser diode hitting on the tilted laser crystal of the micro halfsymmetric cavity created an off-axis strongly asymmetric pumping beam shape that resulted in a half-elliptical gain shape. In such pumping situations, the IGe p,0 and IGo p,1 modes are more efficiently excited than IGMs of Categories 1 and 2.
5. Summary and Discussion of gain-selective IGM excitation
The approaches proposed in this paper to excite all IGMs in an end-pumped solid-state laser are all based on the “mode-gain control” concept. This concept allows any specified IGM to be the mode that is most efficiently pumped in the cavity. The “mode-gain control” concept includes two items, i.e., gain control and loss control. Two examples of gain control are exciting Category 1 IGMs, the IGe p,m modes (p≥m>0), and Category 3 IGMs, the IGe p,0 and IGo p,1 modes. The “gain region control” mechanism was adopted to allow the effective gain region to match the target region of the specified IGMs. Thus, the specified IGM will be the most efficiently pumped mode. An end-pumped solid-state laser system with off-axis pumping and azimuthal pumping could build conditions to excite Category 1 and 3 IGMs separately. A example of both gain control and loss control is the approach to excite the Category 2 IGMs, the IGo p,m modes (p≥m>1). Two mechanisms, “gain region control” and “opaque bar insert,” are used. The “gain region control” mechanism allows the specified IGo p,m mode (p≥m>1) to have a high round-trip gain, and the “opaque bar insert” mechanism allows the gain-competing IGe p,m modes (p≥m>0) to have a much higher loss than the specified IGo p,m mode (p≥m>1). With such proposed “mode-gain control” mechanism, any specified IGMs can be successfully excited in laser systems, as Section 4 demonstrates.
Figure 10 shows how the “mode-gain control” mechanism successfully selects a specified IGM of the three 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 an image source at the point symmetric to the optical axis of laser cavity. Figure 10(b) shows that, different from Category 1 and 3 IGMs, the transient optical source at the gain region of the IGo p,m mode (p≥m>1) and its image source symmetric to the origin further create another two image sources symmetric to the tiny opaque bar situated along the X-axis. 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 IGM output when the round-trip gain of the mode equals the round-trip loss. Figure 11 are movies show detail mode convergence of the three categories, respectively (one frame per ten round trips, 2 frames per second).
In all simulation results, the resulting IGM patterns always match the effective gain regions. Controlling the pump region/effective gain region allows any specified IGM to be selected in an end-pumped solid-state laser system. As Fig. 3 and Fig. 12 show, the target spots of higher-order IGMs are smaller than in lower-order IGMs (the window sizes of sub-figures in Fig. 3 and Fig. 12 are in same scale). A smaller pump region/effective gain region can excite a higher-order IGM. In addition, note that a pump region/effective gain region overlaps many different IGMs with different order p, degree m, and elliptic parameter ε, since ellipticity parameter ε is a continuous, and not discrete, value. The final resulting IGM is the mode whose mode pattern best matches the gain region. Besides, differences between the pump region/effective gain regions of higher-order IGMs are much smaller than in lower-order IGMs. This leads to that a small change of the effective gain region in only sub-micrometer scale will causes the laser oscillation to converge to a different IGM. This sensitivity property of IGM formation might be used to design pattern-reorganization optical switches.
The proposed mode-gain control mechanism has complementary advantages and disadvantages compared to other potential subsequent mode conversion methods to IGMs excitation, i.e., passing the laser beam through some optical element such as a hologram, liquid crystal, or phase plate. The subsequent mode conversion methods can produce IGMs with definite order p, degree m and ellipticity parameter ε. The present numerical study shows pump-beam focusing conditions for forcing the lasers to any desired IGM of all kinds, which also are demanded by and beneficial to the future exploration of the dynamic properties of such novel IGMs and further laser applications.
On the other hand, the proposed method seems to have one drawback: the “mode-gain control” mechanism can exactly choose the order p and degree m of the selective IGMs in an end-pumped solid-state laser system, but it can only roughly estimate the elliptic parameter ε of selective IGMs in advance. Though it is hard to find a definite effective gain region to an IGM with specified ellipticity parameter ε, one quantitative value could estimate an IGM of which ellipticity parameter ε will be excited with a given gain distribution. The quantitative value is matching factor, MF, the correlation coefficient between effective gain regions and resulting IGM’s distribution:
where g is the normalized effective gain distribution function, and upm is the normalized IGM field distribution function. The matching factor, MF, can estimate how the effective gain region matches an IGM. An IGM with a highest value of matching factor, MF, to a specified effective gain region is expected to be the resulting oscillation mode in the end-pumped solid-state system.
In order to verify this idea, we examined the relation between matching factor and the IGM to be excited numerically. Example results for IGe p,m modes are shown in Fig. 13. The red elliptical lines shows the effective gain region for IGe 6,2 mode excitation in simulation, which is addressed in the Table 1.
Four analytical IGM patterns Figs. 13 (d), (b), (e), and (f) shown horizontally are analytical solutions of IGe 6,2 modes with different ellipticity parameters ε. While, the numerical simulation results in the pattern (b) possessing the largest MF value as demonstrated in Fig. 5(b). This implies that the given effective gain region to excite IGe 6,2 mode addressed in the Table 1 will result in IGe 6,2 mode with ellipticity parameter ε close to 1, although the exact ellipticity parameter ε of resulting IGe 6,2 mode to a given gain region could be further solved by maximization of the function MF (ε) .
Nevertheless, the matching factor, MF, can be used to judge IGM of which order p and degree m will be excited to a given gain region in advance. As for three analytical IGM patterns shown in Figs. 13 (a), (b), and (c) vertically, in which the given gain region overlaps all three IGM field distributions, the IGM with a highest value of MF, i.e., IGe 6,2 mode, will be the resulting oscillation mode,
This paper presents a first systematic approach to excite all Ince-Gaussian modes in an end-pumped solid-state laser system. The proposed Ince-Gaussian mode excitation mechanism is based on the “mode-gain control” concept. All IGMs are divided into three categories according their beam characteristics, and the scheme to excite each Ince-Gaussian mode category is explored and verified by numerical simulation. This study also provides numerical evidence for Ince-Gaussian modes observation experiments in half-symmetric-cavity solid-state lasers with system-asymmetry control [7–9]. The proposed approach for producing any specified IGM in a real laser system is important, and will be beneficial to further study on the properties and potential applications of Ince-Gaussian modes.
This work was supported in part by a grant from the National Science Council of Taiwan, R.O.C., under contract no. NSC 96-2112-M-006-019-MY3
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