Volume 7 Issue 8 - February 27, 2009
Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers
Shu-Chun Chu

Department of Physics, College of Sciences, National Cheng Kung University
scchu@mail.ncku.edu.tw

OPTICS EXPRESS 15, pp. 16506-16519 (10 December, 2007)

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Researchers have recently predicted a third complete family of transverse modes, PWE solutions in elliptic cylindrical coordinates, namely the Ince-Gaussian modes (IGMs), which constitute the continuous transition modes between Hermite-Gaussian modes (HGMs) and Laguerre-Gaussian modes (LGMs). IGMs are recently observed in a LD-pumped solid-state laser by breaking the symmetry of the cavity. These attempts include introducing a cross hair inside the cavity or adjusting the azimuthal symmetry of the short laser resonator. However, a general rule for exciting any specified IGM in a laser system has yet to be discovered. This study reports 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.

This study 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, which simulates a single-wavelength, single/multi-mode oscillation in unstable/stable laser cavities. As Fig. 1 shows, this study models a half-symmetric laser resonator similar to that in real experiments. This cavity is formed by one planar mirror and a concaved mirror with a curvature radius of R2 = 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.
Fig. 1. Diagram of the simulation model of a half-symmetric cavity

This study classifies IGMs into three categories according to their characteristics, which lead to different resulting mechanisms. The three categories are: (1) IGep,m modes (p≥m>0), (2) IGop,m modes (p≥m>1) and (3) IGep,0 and IGop,1 modes. Figures 2 (a), (b), and (c) show some typical patterns of the three categories of the IGMs, respectively. 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,om,m mode is actually the inner pattern of IGe,op,m mode with parameter p>m. However, these IGMs are separated into two categories because only the IGop,m mode patterns break along the X-axis–the IGep,m modes do not. The following summarizes the scheme to excite each category of IGMs.

Fig. 2. Analytical amplitude distribution of the three categories of IGMs: (a) IGep,m modes (p≥m>0), (b) IGop,m modes (p≥m>1) and (c) IGep,0 and IGop,1 modes.

All the approaches to IGMs excitation in real experiment introduce the change of the effective gain region size and transverse position at the laser crystal. Figure 3 shows the proposed corresponding effective gain region to excite each category of IGMs.
Fig. 3. Effective gain region at laser crystal to excite (a) IGep,m modes with p≥m>0 (b) IGop,m modes with p≥m>1 and (c) IGep,0 and IGop,1 modes

(1)Exciting IGep,m modes (p≥m>0): As the red elliptic lines plotted in Fig. (a) indicate, the pattern focus is located at the brightest (i.e., target) spot of the IGep,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 IGep,m modes with p≥m>0 in the end-pumped solid-state lasers 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.

(2)Exciting IGop,m modes (p≥m>1): The pattern of IGop,m modes differs from IGep,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 IGep,m modes will suffer significant energy loss in one round-trip of the cavity due to diffraction by the tiny opaque bar; however, IGop,m modes will not. This means that using both the “gain region control” and “opaque bar insert” mechanisms together, any IGop,m mode (p≥m>1) can be excited in an end-pumped solid-state laser system.

(3)Exciting IGep,0 and IGop,1 modes: The IGep,0 and IGop,1 modes have a similar mode distribution that-- (a) most pattern spots are in half-elliptical form, (b) 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.


Figure 4 shows simulation progress of how the “mode-gain control” mechanism successfully selects a specified IGM of the three categories from an initial random field pattern. At first, the initial random field contains all resonator eigenmodes. Our approach is simply by adjusting the effective gain region size and transverse position at the laser crystal to result in the specified IGMs is the mode of highest round-trip gain. Thus, after several traveling back and forth in the resonator, only the specified IGM will survive and all other modes will die out.
Fig. 4. Demonstration of resulting progress of stable amplitude distribution of (a) IGep,m modes (p≥m>0), (b) IGop,m modes (p≥m>1) and (c) IGep,0 modes  and IGop,1 modes. The left-hand side images are simulated spontaneous emissions with partially coherent random fields.

In summary, this study 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. The proposed approach for producing any specified IGM in a real laser system will be beneficial to further study on the properties and potential applications of Ince-Gaussian modes.
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