Various single-frequency Ince-Gaussian mode oscillations have been achieved in laser-diode-pumped microchip solid-state lasers, including LiNdP4O12 (LNP) and Nd:GdVO4, by adjusting the azimuthal symmetry of the short laser resonator. Ince-Gaussian modes formed by astigmatic pumping have been reproduced by numerical simulation.
©2007 Optical Society of America
The transverse lasing eigenmodes in stable resonators have been investigated with analytical, numerical, and experimental techniques. In addition to the well-known Hermite-Gaussian (HG) modes and Laguerre-Gaussian modes, a third complete family of transverse eigenmodes, Ince-Gaussian (IG) modes, has been predicted as orthogonal solutions of the paraxial wave equation in elliptic coordinates . IG mode operations have been observed experimentally in a laser-diode-pumped Nd:YVO4 laser by breaking the symmetry of a long nearly semiconcentric external cavity with a length of 6.8–29.5 cm, with a slight sideways shift of one concave output coupler, perpendicular to the pump direction by several tens of micrometers, i.e., “off-axis pumping”, and occasionally introducing a crosshair inside the cavity . The mechanism of forced IG mode operations is that the beam diameter of the spot of the higher-order IG modes is smaller than the fundamental mode diameter, so IG modes are more efficiently pumped by the tight pump focus .
In this paper, we report demonstrating a variety of single-frequency IG mode operations in several microchip solid-state lasers with a short cavity configuration, i.e., 5-mm-long semiconfocal resonator, assembled into one body. Forced IG mode operations were achieved easily by tilting the central axis of the resonator with respect to the axis of the laser diode (LD) pump beam without using an intracavity crosshair. The changes in lasing pattern with pump power and tight pump focus position were demonstrated experimentally and reproduced by numerical simulation of IG mode formation in a microchip short-cavity resonator subjected to astigmatic pumping.
2. Experimental setup and forced IG mode operations
2.1 LNP laser
The experimental setup is shown in Fig. 1(a). The first experiment was carried out using a 1- mm-thick a-plate LiNdP4O12 (LNP) crystal, whose absorption coefficient for the LD wavelength of 808 nm is 115 cm-1. The resultant absorption length was as short as 87 µm. An elliptical LD beam was transformed into a circular one and focused onto the LNP crystal by using a microscope objective lens with a numerical aperture of 0.25 to obtain a tight focus giving a minimum spot size of approximately 75 µm at the crystal. The LNP crystal was placed within a semiconfocal external cavity, in which the sample was attached to a plane mirror M1 (99.8% reflective at 1064 nm and >95% transmissive at 808 nm) and a concave mirror M2 (99% reflective at 1064 nm, radius of curvature: 10 mm) was placed 5 mm away from the plane mirror. The two mirrors and the LNP crystal were assembled into one body.
By adjusting the azimuthal symmetry, we achieved a variety of higher-order IG mode oscillations, IGp,m, where the central axis of the resonator was tilted with respect to the pumpbeam axis, as shown in Fig. 1(a), in which the tilt angle was changed in the range of 0<θ< 30 [mrad]. Such a tilt of the integrated laser resonator is considered to introduce an effect equivalent to off-axis pumping with a lateral shift of 0<d<150 µm , since the pump light is absorbed within 100 µm. In the absence of a tilt, i.e., astigmatic pumping, the laser exhibited an HG0,0 mode (i.e., TEM00 mode) oscillation, where the threshold pump power was 140 mW and the slope efficiency was 8%. The relatively high threshold pump power for a 1- mm-thick LNP crystal results mainly from the large resonant reabsorption loss in the non-pumped region in the stoichiometric LNP lasers with high Nd density . Examples of far-field lasing patterns observed for different azimuthal symmetries at a constant pump power of 293 mW are shown in Fig. 1(b), where the output powers were 30–50% lower than those for TEM00 mode operation. Linearly polarized emissions along the pseudoorthorhombic c-axis were observed. Here, a crosshair was not needed to achieve these IG modes.
Mode numbers [p,m] and ellipticity ε of the mode were determined from the correspondence to patterns calculated analytically using the following equation.
Here, C is a normalization constant, w(z) is the beam width [w 0=w(0)], the superscript e refers to even modes, and Cm p(•, ε) are the even Ince polynomials of order p, degree m, and parameter ε. The elliptic coordinates were defined as x=f(z) cosh ε cos η, y=f(z) sinh ε sin η, and z=z, where f(z)=f0 w(z)/w 0 is the semifocal separation and ξ and η are the radial and angular elliptic variables, respectively. We obtained odd IGop,m(r, ε) by writing the odd Ince polynomials Smp(•, ε) and odd normalization constant S instead of the even ones. r is the radial distance from the central axis of the cavity, R(z)=z+z2R/z is the radius of curvature of the phase front, and Ψz(z)=arctan(z/zR). IG mode patterns calculated by Eq. (1), which correspond to observed patterns in Fig. 1(b), are shown in Fig. 1(c).
In the case of the Nd:YVO4 laser in Ref. , the fundamental mode has been reported to switch to higher-order IG modes with increasing pump power after adjustment of the azimuthal symmetry, under a tight pump beam focusing condition . As for the present short-cavity LNP, on the other hand, the larger thermal lens effect due to the low thermal conductivity of LNP (i.e., K=3.2 [W/mK]) was found to modify the resonator configuration with increasing pump power and resulted in pump-dependent lasing pattern changes. Three typical types of structural changes are shown in Fig. 2 where the azimuthal symmetry was fixed constant for each pattern change. In the case of Fig. 2(a), the pattern switched from lower- to higher-order IG mode with increasing pump power, similar to Ref. , while the IG mode appeared as a structural change resembling the HG mode, in Fig. 2(b). IG-mode formation featuring a change in symmetry was also observed, as shown in Figs. 2(c) and 2(d), where IG modes were formed from two-fold symmetric lasing patterns shown in the left column, leading to drastically different two-fold symmetric lasing patterns possessing two planes of symmetry indicated by dotted lines shown in the right column.
It is worthwhile to point out that selective excitations of higher-order IG modes in LNP lasers result in a pronounced lasing-frequency selectivity compared with TEM00 mode operations. In LNP lasers, simultaneous operations on different transitions, 4F3/2(1)→4I11/2(1,2,3), at wavelengths of 1048, 1055, and 1060 nm, take place due to transition-dependent resonant reabsorption losses inherent in stoichiometric lasers having high Nd densities [3,4].
Figure 3 shows modal input-output characteristics and optical spectra measured with a multi-wavelength meter of a resolution of 0.1 nm in TEM00 mode operations with θ=0, in which example far-field patterns are also shown. In short, lasing occurred in the sequence of 1048-nm→1060-nm→1055-nm transitions with increasing pump power . When higher-order IG modes were forced to oscillate, on the other hand, lasing on only the 1048-nm transition was observed independently of the tilt θ and pump power. Example results are shown in Fig. 4.
The phase velocity increased with increasing order number, and the Gouy phase shift of the IG modes is given by ΨIG=(p+1)Ψz(z). In short, the transition-wavelength-dependent Gouy phase shift for the lowest order mode Ψz(z) in LNP lasers was enhanced in the Gouy shift for higher-order modes by a factor (p+1), which determined the lasing frequency of IGp,m modes. This implies that when the higher-order IG mode pattern belonging to the 1048-nm transition possessing the highest gain [3,4] was formed by tuning θ, lasing in the same IG mode pattern could not take place at other transitions because of different Gouy shifts. Moreover, the 1048-nm transition was identified as supporting a single longitudinal mode in IG-mode operations by measuring detailed optical spectra with a scanning Fabry-Perot interferometer with a 2-GHz free spectral range and a resolution of 6.6 MHz. Example results indicating single-frequency IG-mode oscillations are shown in Fig. 5.
2.2 Nd:GdVO4 laser
The second experiment was performed by replacing the LNP crystal by a 1-mm-thick a-cut Nd:GdVO4 crystal with 3 at.% Nd in the same resonator. The absorption coefficient for the LD pump light was 74 cm-1. Due to the resultant short absorption length of 135 µm, a tilt of the resonator induced the effect of precise off-axis pumping for generating IG modes, similar to the case of LNP lasers. The threshold pump power in the absence of cavity tilt was greatly reduced to 20 mW and the slope efficiency was 13% because of the larger emission cross section of Nd:GdVO4, i.e., σe=7.6×10-19 cm2, and the negligible reabsorption loss, while σe=1.7×10-19 cm2 for LNP.
Various IG modes were observed by changing the tilt of the short laser-cavity, similar to the LNP laser. In all cases, single-frequency, linearly π-polarized emissions along the tetragonal c-axis were observed. Example patterns are shown in Fig. 6(a). A successive structural change to IG modes was also observed by changing the cavity position in the pump direction (z-axis), as shown in Fig. 6(b), where the pump beam diameter inside the Nd:GdVO4 crystal was changed. In the case of Fig. 6(b), higher-order IG modes were formed with decreasing pump beam diameter, i.e., tight pump focus, as will be shown when the simulations are discussed. IG mode patterns calculated by Eq. (1), which correspond to observed patterns in Figs. 6(a) and 6(b), are shown in Fig. 6(c).
Like the HG and LG modes, the IG modes are expected to exhibit shape invariance. Figure 7 shows examples of the far-field patterns of the Nd:GdVO4 laser observed at different propagation planes. These corroborate the shape invariance against propagation. The near-field pattern had the same shape.
Figure 8 shows an example higher-order IG mode observed in the high pump-power region and its structural change versus the change in the cavity (i.e., pump) position in the y and z directions. Corresponding analytically calculated IG patterns are also shown on the right of Fig. 8. In the case of Nd:GdVO4 lasers, such higher-order IG mode patterns were maintained in a wider pump-power region than for LNP lasers, presumably due to the smaller cavity deformation resulting from the thermal lens effect. The effective focal length of a thermal lens is proportional to K/[(dn/dT)+α(n-1)], where dn/dT is the coefficient of thermal refractive index change, n is the refractive index, and α is the coefficient of thermal expansion . Here, the thermal denominator value is almost the same for both lasers, i.e., 6.2×10-6/K (Nd:GdVO4)  and 6.3×10-6/K (LNP) . Since the thermal conductivity of Nd:GdVO4 crystal, K=11.7 [W/mK], is about four times that of LNP crystals, a smaller thermal lens effect is expected in Nd:GdVO4 lasers. When the pump power was increased further, however, IG mode patterns changed: multiple IG mode operations appeared, as shown in Fig. 8.
3. Numerical simulation of forced IG mode operations by Azimuthal pumping
In this section, we numerically demonstrate how a single IG mode pattern was chosen by using azimuthal tight focus pumping. The changes in lasing pattern with pump power and tight pump focus position were checked by numerical simulation of IG mode formation in a microchip short-cavity resonator subjected to astigmatic pumping.
The simulation code we used was created based on Endo’s simulation method [8–10], which could simulate a single-wavelength, single/multi-mode oscillation in an unstable/stable laser cavity [8–10]. We summarize the method as follows. The method simulates the initial stimulated field with a partially coherent field  in the space-frequency domain to avoid the 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 Fresnel-Kirchhoff integration . Optical fields that are changed by laser mirrors and the gain medium are introduced by modifying the optical field at each position. In this model, the modification of the laser mirrors and thermal lens can be easily represented by a phase change . The loaded gain at each station is assumed to be homogeneously broadened. The gain medium was treated as several gain sheets. The saturated gain at each gain sheet i is expressed as
where g0(x,y) is the loaded gain, gi0(x,y) is the small signal gain, and Is(x,y) is the saturation intensity. Here, we assumed Is≈1 kW/cm2 , which is appropriate for solid-state lasers. The symbol Ĩ+ i and Ĩ- i are the average right-going and left-going optical intensities, which were defined as
The I + i(q) and I + i(q) denote the intensities of the qth iteration step, and α is a summation taken over the period of the photon decay time of the cavity.
For example, in the cavity used in our experiment, the reflectivities of the two mirrors were r1=99.8% and r2=99%, respectively. Thus, the parameter α is given by r1*r2=0.988. 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, the cavity finds the lasing mode distribution E(x, y) that satisfies
We checked the validity of the code we made by repeating simulations of a single-mode oscillation of the unstable resonator and a multimode oscillation of the stable resonator in Ref. . After that, we used the code to study azimuthal tight focus pumping in our microchip laser system.
The laser cavity configuration assumed for simulations is illustrated in Fig. 9, where the gain region was assumed to be localized near the pumped crystal surface corresponding to short absorption lengths for the LD pump beam in both laser crystals used in the experiment. The gain region (‘cylinder’) size within the crystals was set to be 50 µm in radius and 100 µm in thickness. Here, we assumed that the radius of the excess gain region over the loss was smaller than the spot size of the focused LD beam not far above the threshold pump power. When the gain region was located at the central axis of the cavity, i.e., the lasing axis, the HG mode was realized as a stationary lasing pattern after iterations. When the gain region corresponding to the pumped focus was shifted laterally by d=60 µm, as depicted in the inset of Fig. 10, a single IGe2,2 mode oscillation was numerically found to be always established as a stationary lasing pattern, starting from random initial patterns with different internal power variations. If no mode competition occurred, the mode converged rapidly. The lasing pattern converged slowly when mode competition occurred during the iterations, but finally a single IG mode, which is most efficiently pumped, survived. An example result is shown in Fig. 10, where the relative internal power, which represents the internal power ratio of a transient pattern to the final pattern, is also shown as a function of the iteration number. The final stationary lasing pattern was formed after 500 iterations in this case.
When the lateral shift of the gain region was increased to d=75 µm, a single IGe3,3 mode lasing pattern was formed from a random pattern, as shown in Fig. 11(a). The numerical results shown in Figs. 10 and 11(a) imply that the IG mode field, which is most effectively amplified by a shifted pump focus through a match between a partial lasing spot and a pump focus, is realized as a stationary lasing pattern, being selected from transverse eigenmodes after many iterations. In the actual experiment, adjustments of the relative gain position and gain region size in the microchip laser were introduced by a cavity rotation (θ) and lateral and longitudinal translations of the microchip cavity (Δy, Δz), as demonstrated in the experiment.
In addition to the selection of the relative gain position in the laser cavity after adjustment of the azimuthal symmetry of the microchip cavity, a change in the pumping power is another important factor in selecting the lasing IG mode. With increasing pumping power, both the effective gain region and the small signal gain increase. Assuming an increased gain region with a radius/thickness of 70/150 µm and a doubled small signal gain, we can see a structural change in the lasing pattern from the IGe3,3 mode back to the IGe2,2, mode, which is most effectively amplified by the increased gain region, as shown in Fig. 11(b). This implies that the lasing mode is sensitive not only to the relative gain region with respect to the lasing axis, but also to the pump power, as demonstrated in the experiment. An effect similar to a change in the lateral gain size on structural pattern changes was observed in the experiment, as shown in Fig. 6(b).
4. Mixed-mode operations-
When the crystal was rotated around the cavity axis, where the polarization direction of the LD pump light was not parallel to the crystal c-axis, i.e., the polarization direction of the solid-state lasers, lasing patterns consisting of superimposed transverse modes appeared. Results obtained with the Nd:GdVO4 laser are shown in Figs. 12 and 13. As shown in Fig. 12, the laser tended to support two patterns whose symmetric axes were nearly perpendicular, where these two patterns appeared as a result of a slight shift of the resonator in the pump direction, i.e., a change in the lateral gain area. The anisotropic absorption of the pump light along the b- and c-axes may also result in anisotropy of the gain distribution in the longitudinal direction. As the rotation angle and the azimuthal symmetry were tuned, mixed modes possessing four-fold symmetries were observed with increasing pump power. An example of the pump-dependent pattern change is shown in Fig. 13(a). The starting pattern in Fig. 13(a) is given analytically by a superposition of two IGe2,0 modes, as shown in Fig. 13(b), where the ellipticity dependence was not identified. These results suggest that the strong anisotropic property in Nd:GdVO4 crystal affects the formation of lasing patterns in symmetry-broken optical resonators, resulting in curious symmetric patterns consisting of mixed global solutions of optical resonators, i.e., HG, LG, and IG modes . In such mixed-mode operations, scanning Fabry-Perot interferometer traces exhibited multiple frequency peaks.
We have demonstrated forced Ince-Gaussian (IG) mode operations in microchip solid-state lasers with laser-diode azimuthal pumping, using stoichiometric LiNdP4O12 and 3 at.% Nd-doped Nd:GdVO4 crystals. Short absorption lengths for LD pump light enabled us to perform off-axis pumping by tilting the microchip laser cavity, which was assembled into one body, with respect to the pump beam axis. With precise adjustments of the effective gain volume and a shift of the gain region from the central axis of the cavity caused by the pump power, cavity tilt and lateral and longitudinal translations of the laser crystal, various single IG mode lasing patterns were formed without an intra-cavity crosshair being used. All the observed patterns were successfully reproduced analytically.
The formation of IG modes in the present microchip solid-state lasers with simple azimuthal pumping was well reproduced numerically. A pronounced lasing frequency selectivity associated with forced IG mode operations resulting in single-frequency oscillations was demonstrated experimentally and discussed in terms of pronounced Gouy phase shifts inherent in higher-order IG modes. Curious mixed transverse lasing mode operations were also demonstrated in the high pump-power region.
KO is indebted to Professor J.-Y. Ko and C.-C. Lin of National Kaohsiung Normal University, Taiwan, for discussions.
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