## Abstract

A straightforward method is presented for generating a stigmatic spherical thermal lens in laser-diode-pumped, Brewster-cut solid-state gain media by shaping the aspect ratio of the elliptical pumped region. Demonstration of this laser head design with Nd:GdVO_{4} as the gain medium yields a stable, efficient, high-power (>20W) diode-pumped laser at 1063nm. Analysis of the spatial mode characteristics of a 67cm-long symmetric resonator both confirms the radially symmetric nature of the pump-induced thermal lens and indicates that laser resonators incorporating this head design can readily generate a high spatial beam quality (*M*
^{2}<2).

©2004 Optical Society of America

## 1. Introduction

The directionality and spectral purity (i.e., brightness) of laser diode (LD) pump sources has revolutionized solid-state laser development [1,2]. In particular, many diode-pumped solid-state lasers producing multiwatt continuous-wave (cw) output power in the fundamental TEM00 spatial mode employ fiber-coupled LD pumping to provide the high efficiency generated by good spatial overlap between the pump radiation and the intracavity laser mode [3–5]. An optimized end-pumped Nd:YVO_{4} laser of this type has produced 25W of cw output power in the fundamental TEM_{00} mode at 1064nm [6]. Other efficient LD-pumped laser architectures have also been developed: for example, the thin disk geometry [7] as well as self-adaptive phase-conjugate oscillators [8] based on the grazing-incidence geometry [9]. While the former has been shown to be capable of generating kW-level cw output power using Yb:YAG as the gain medium [10], the latter ingeniously compensates for the strong pump-induced thermal lensing in Nd:YVO_{4} to produce 20W of cw TEM_{00}-mode output. Several commercial devices based on fiber-coupled end-pumped and, more recently, thin disk [11,12] geometries are now available.

In this paper, we outline an alternative technique for obtaining high-power, low-divergence output from axially LD-pumped solid-state gain media. The method exploits the directionality and output asymmetry of LD arrays to generate a stigmatic spherical (i.e., radially symmetric) thermal lens in inherently astigmatic Brewster-cut laser crystals by shaping the aspect ratio of the elliptical pumped region. The rationale behind this straightforward method is described in Section 2. In Section 3, we propose a laser head design, which also avoids the use of dichroic or anti-reflection (AR) dielectric coatings, to test the concept using Nd:GdVO_{4} directly pumped by two 40W LDs in a near-longitudinal counter-propagating geometry. The cw performance of this high-power Nd:GdVO_{4} laser head in symmetric flat-flat resonators is described in Section 4. In particular, results of a spatial mode analysis are presented to verify that the design goal of shaping a stigmatic spherical thermal lens in the Brewster-cut gain medium has been achieved.

## 2. Design rationale

The use of Brewster-cut solid-state gain media in laser resonators has a number of well-known advantages: (i) the near-zero intracavity reflection loss from the gain medium improves laser efficiency, (ii) the geometry strongly favors cavity oscillation in the linear π-polarization state (even if the gain medium is optically isotropic like Nd:YAG), and (iii) the optical damage threshold is generally higher for π-polarized radiation incident at Brewster’s angle than for normal incidence radiation (especially if an AR dielectric coating is used). However, the inherent astigmatism of the Brewster geometry will generally result in output beams with elliptical spatial profiles, unless optical compensation schemes (e.g., tilted focusing optics) are employed [13]. Below, we propose an alternative astigmatism compensation scheme for axially LD-pumped Brewster-cut solid-state gain media in which the pump-induced thermal lens is shaped to present, in effect, a stigmatic focusing element to the laser resonator.

Consider the longitudinally LD-pumped, Brewster-cut solid-state gain medium shown in Fig. 1. The inherent radial asymmetry of the Brewster geometry will, in general, produce an astigmatic pump-induced thermal lens due to an elliptical temperature profile that, in the parabolic paraxial approximation, has the form

Here, we have used the coordinate system defined in Fig. 1, so that *T*
_{0}=*T*(*x*=0, *y*=0) is the peak axial temperature, and the values of *A*_{T}
and *B*_{T}
are determined primarily by the spatial pump beam profile and the details of thermal conduction in the tangential (*x*-*z*) and sagittal (*y*-*z*) planes, respectively. Normally, *A*_{T}
and *B*_{T}
are also functions of *z* (the axial co-ordinate) due to pump beam absorption (and possible self-induced focusing [14]) over the crystal length *l*. However, for the counter-propagating pump geometry employed for the Nd:GdVO_{4} laser described below, where the absorption-length product *αl*≈2, *A*_{T}
and *B*_{T}
may be approximated as constants.

The spatial temperature profile of Eq. (1) generates three contributions to the overall thermal lens of the gain medium [2]: (i) an astigmatic gradient index (GRIN) lens through the bulk thermo-optic effect, (*dn*/*dT*); (ii) a refractive lens caused by bowing of the two faces of the gain medium due to thermal expansion; and (iii) a generally nonsymmetric lens produced by stress-induced changes in the refractive index. The stress contribution, which causes birefringence in Nd:YAG rods [2], is difficult to separate and quantify in birefringent gain media like Nd:GdVO_{4} and Nd:YVO_{4} that have a strong preferential laser emission polarized parallel to the crystal c-axis. Moreover, as shown later, this effect is expected to be insignificant for our pumping geometry. On the other hand, the thermal GRIN lens over the gain medium length *l* and the bowing of the crystal faces are both major contributors – albeit with a net thermal focal length *f*_{T}
≫*l*.

The optical effects of both the GRIN and refractive bowing contributions to the thermal lens in the Brewster geometry of Fig. 1 can be represented by ray transfer matrices (RTMs) [15]. For our case, where the GRIN lens’s perturbation to the wavefront is small upon propagation through the gain medium with refractive index *n*, the net RTM in the tangential plane (*x*) is

and that in the sagittal plane (*y*) is

In Eqs. (2), different GRIN lens coefficients *γ*_{i}
and implicit positive bowing radii of curvature *R*_{i}
(*i*=*x* or *y*) are assumed in the sagittal and tangential planes due to the elliptical thermal spatial profile of Eq. (1). The constants Δ*y* and Δ*x* are given by

where θ
_{B}
=tan^{-1}(*n*) is the Brewster (or polarization) angle.

It is clear from *ABCD* matrices of Eqs. (2) that in the ‘thin lens’ approximation (*f*_{T}
≫*l*), which is valid in our case, only the terms in the square brackets will contribute predominantly to the thermal lens. Consequently, since the focal length associated with a thin optical system described by an *ABCD* matrix is given by the element ratio -(*A*/*C*), the condition for a Brewster-cut gain medium to present a stigmatic thermal lens may be written as

In obtaining Eq. (4), we have related the GRIN lens coefficients to the constants *A*_{T}
and *B*_{T}
defined by Eq. (1): ${\gamma}_{x}^{2}={A}_{T}\left(\frac{\mathit{dn}}{\mathit{dT}}\right)$ and ${\gamma}_{y}^{2}={B}_{T}\left(\frac{\mathit{dn}}{\mathit{dT}}\right)$. In addition, we have computed the values of *R*_{i}
by assuming that only parts of the gain medium within a depth *d* of the surface contribute to the bowing through the thermal expansion coefficient α
_{T}
. In a uniformly-pumped Nd:YAG rod, this distance is roughly equal to the rod radius [2]; that is, half of the spatial width of the pumped region. The corresponding distance in a ‘softer’ material like Nd:glass is approximately equal to the entire spatial width of the pumped region (i.e., the diameter for a uniformly-pumped rod). For the Nd-doped GdVO_{4} crystal, the characteristic expansion distance *d* is expected to lie between that of crystalline YAG and amorphous glass. This expansion distance must also be the same in both the tangential and sagittal planes, irrespective of the pumping geometry, in order to minimize the net internal stress and strain in the gain medium.

Equation (4) clearly indicates that, for a laser crystal with given dimensions and properties, there should always exist an elliptic parabolic temperature distribution (i.e., ratio *A*_{T}
/*B*_{T}
) within the thin lens approximation for which a Brewster-cut gain medium has a stigmatic thermal lens. Below we describe a general means of achieving this condition by manipulating the spatial pump beam profile in a laser-head geometry that restricts thermal conduction to one dimension [16,17]. The validity of this design approach is verified using results from a high-power (>10W), diode-pumped Nd:GdVO_{4} laser with high spatial beam quality.

## 3. The diode-pumped Nd:GdVO_{4} laser head

To affect the aspect ratio of the tangential-sagittal astigmatism of a thermal lens in a Brewster-cut solid-state laser gain medium, it is generally necessary to regulate the thermal conduction geometry and to closely control the spatial dimensions of the pumped region. The advent of highly directed pump radiation from LDs has made the latter control mechanism possible. To simplify the manipulation of the thermal lens, we have chosen to restrict the thermal conduction to one dimension (the sagittal (*y*) direction) so that the spatial extent and shape of the pump radiation in the perpendicular direction (the tangential (*x*) plane) solely determines the strength of the thermal lens in that dimension.

The diode-pumped, Brewster-cut Nd:GdVO_{4} laser head that we have developed to manipulate the thermal lens is depicted in Fig. 2. The 0.5at.%-doped, 3×5×10mm Nd:GdVO_{4} laser crystal is appropriately cut for Brewster-angled *c*-axis emission through its polished 3×10mm faces. To secure one-dimensional heat removal [16,17], only the largest 5×10mm top and bottom faces are conductively contacted to two water-cooled brass mounts, using silver paste with a thermal conductivity of 11Wm^{-1}K^{-1} [18]. The gain medium is pumped through both polished sides by two 40W LD bars whose outputs at 808nm are first partially fast-axis collimated in the sagittal plane by a cylindrical lens (*f*=10mm) and then focused by a spherical lens (*f*=20mm). To avoid the use of dichroic mirrors (or coatings) and to allow spatial separation from the intracavity laser beam (incident at Brewster’s angle, θB=65.5°), the pump radiation is incident at angles of 15–45° on the gain medium (Fig. 3), which is located at the focal plane of the spherical pump lenses. Nonetheless, the resulting ~15° internal angular separation between the laser and counter-propagating pump beams in the tangential plane creates an efficient quasi-longitudinal pumping geometry over the pumped *l*≈5mm Nd:GdVO_{4} crystal length, the ±600–700µm spatial walk-off being less than half the ~2mm tangential laser beam diameter encountered inside the gain medium of the operational laser (see Section 4). In addition, two half-wave plates are employed to ensure that the 808nm pump radiation from each 40W bar can be rotated into the s-polarization. The use of a pump polarization parallel to the *a*-axis of the 0.5at.% Nd-doped gain medium allows access to a lower α=4cm^{-1} effective pump absorption coefficient [3,19], which (i) is beneficial for high-power diode-pumped Nd:GdVO_{4} lasers [5] and (ii) provides for a more favorable uniform thermal load along the crystal length [20] owing to the resulting α*l*≈2 absorption condition for the counter-propagating pump beams.

The important design feature of the laser head is that the position of the cylindrical lens (L2) with respect to the LD controls the aspect ratio of the elliptical pump beams in the Nd:GdVO_{4} gain medium. Specifically, the position of the cylindrical lens determines the width of the pumped region in the sagittal plane, while that in the tangential plane is held constant by the static spherical lens (L1). This means that the strength of the pump-induced thermal lens in the sagittal plane can be manipulated independently to realize the condition for a stigmatic thermal lens defined by Eq. (4). For the Brewster-cut geometry of Fig. 2, the relevant properties of Nd:GdVO_{4} (*n*=*n*_{e}
=2.198 [21], $\frac{\mathit{dn}}{\mathit{dT}}=\frac{{\mathit{dn}}_{e}}{\mathit{dT}}\approx 4\times {10}^{-6}{K}^{-1}$ [3,22] and *$\overline{\alpha}$
T*≈1.6×10^{-6}K^{-1}, since *α*_{o}
=1.5×10^{-6}K^{-1} and *α*_{e}
=7.3×10^{-6}K^{-1} in the ordinary (*a*-axis) and extraordinary (c-axis) directions, respectively [3]) predict that Eq. (4) is satisfied for a ratio *A*_{T}
/*B*_{T}
≈0.19; i.e., the thermal lens in the sagittal plane should be about five times stronger than that in the tangential plane. We note that the same calculation for Nd:YAG (*n*=1.82, $\frac{\mathit{dn}}{\mathit{dT}}=7.3\times {10}^{-6}{K}^{-1}$, and *α*_{T}
=7.9×10^{-6}K^{-1} [2]) gives *A*_{T}
/*B*_{T}
≈0.26, which can also be achieved with the laser head design of Fig. 2.

Figure 3 illustrates results of an exact ray tracing for the pump radiation from one 40W LD array to the Brewster-cut Nd:GdVO_{4} gain medium through a canonical arrangement of the spherical (L1) and cylindrical (L2) pump optics. In the tangential plane (Fig. 3(a)), the pump spot size is determined solely by the focal length of L1 and by the slow-axis divergence of the 46 1×80µm individual semiconductor lasers comprising each 40W diode bar, as the gain medium is positioned at the focal plane of the two spherical lenses. Since the specified full-width divergence is 10° at 1/e^{2} irradiance (FW1/e2I) [23], a quasi-Gaussian pump beam with a FW1/e^{2}I of 4–5mm results in the gain medium, as indicated in Fig. 3(a). This tangential pump spot size is not significantly affected by either the spherical aberrations of the lens L1 or the inherent astigmatism of the Brewster-oriented gain medium. As a result, the parabolic contribution *A*_{T}
to the tangential (*x*) direction of the elliptical temperature distribution (Eq. (1)) is expected to be characterized by the tangential pump spot size, since the thermal conduction is solely in the orthogonal sagittal direction. This is not true in the sagittal plane, where spherical aberrations, predominantly from the cylindrical lens L2, severely perturb the spatial shape of the pump beam. In fact, as shown in Fig. 3(b), pump rays near the specified fast-axis FW1/e^{2}I of 40° [23] (i.e., ±20° from the optical axis) are brought to a focus several millimetres before the central rays. The result is that a pump irradiance profile closer to a ‘top hat’ (or super-Gaussian) is produced in the gain medium. This means that the sagittal thermal lens for the pumped region between the two heat removal planes is more parabolic than in the absence of spherical aberration in the pump optics. The full-width of the sagittal pump spot size can be varied between about 0.7 and 1.5mm, depending on the axial position of the cylindrical lens. The exact value of the parabolic contribution *B*_{T}
to the sagittal (*y*) direction of the elliptical temperature distribution (Eq. (1)) is a function of this pump spot size and the 3mm height of the gain medium. We note that the ratio *A*_{T}
/*B*_{T}
will be independent of the pump power, provided that the thermal conductivity κ is not a strong function of the local Nd:GdVO_{4} crystal temperature.

## 4. Laser performance and characterization

The performance of the diode-pumped, Brewster-cut Nd:GdVO_{4} laser head was assessed using simple plane-plane resonators of varying length with the gain medium (a positive thermal lens) positioned at the symmetric center of the cavity. Figure 4 shows the laser output power at 1063nm as a function of diode drive current (and summed diode pump power) for three different resonators, all with a near optimum 8% output coupling. At 32cm, the shortest cavity length, the 20W output power level at a diode current of 41A corresponds to an overall optical-to-optical efficiency of 32%, or, equivalently, a laser-output to absorbed-pump-power efficiency of ~41%, once reflective losses for the incident pump radiation have been accounted for. The initial optical-to-optical slope efficiency common to all resonators is 37(±2)%, which compares reasonably well with conventional fiber-coupled LD end-pumped geometries [3,5,19] after considering the diminished spatial overlap between the pump radiation and the intracavity laser mode. The reduction in output power for longer cavity lengths at high diode pump powers is due to the proximity of the confocal stability point for the symmetric resonators, which causes an increasing spatial mismatch between the diode-pumped region and the intracavity laser mode in the gain medium – specifically, the lowest-order TEM_{00} laser mode size exceeding the ~1mm pumped region in the sagittal plane (Fig. 3(b)). The position of this boundary, which is indicated by the solid black line in Fig. 4 and distinguishes stability regions I and II of the cavity, is evaluated using data for the power of the thermal lens in the Nd:GdVO_{4} crystal extracted from the analysis of the spatial properties of the laser output presented below. When operated well within stability region I, the diode-pumped Nd:GdVO_{4} laser has exceptional power stability (<1% power drift over one hour) and beam pointing instabilities significantly less than 100µrad, all without any protection (e.g., a housing) from the laboratory environment. Operation in stability region II, and particularly near the confocal ‘edge,’ is less stable, primarily due to observed fluctuations in the laser’s transverse mode structure.

The laser power data displayed in Fig. 4 was obtained under the circumstance that the diode-pumped Nd:GdVO_{4} gain medium presents a radially symmetric thermal lens to the resonators; i.e., the condition defined by Eq. (4) is met. In practice, this condition is determined by finding the position of the cylindrical lens (L2) for which the fundamental TEM_{00} mode present at low output powers smoothly transitions into the TEM_{10*} or ‘donut’ mode as the diode drive current is increased. The ability of the laser cavity to support this mode is a strong indicator of radial symmetry in a cavity that contains an astigmatic Brewster-cut gain medium with a strongly preferred linearly polarized emission. If the cylindrical lenses are offset from this optimum position towards the LD arrays, which results in a larger sagittal pump spot size in the gain medium, the initial TEM_{00} mode transitions into a horizontal TEM_{10} Hermite-Gaussian spatial mode. Conversely, an offset towards the Nd:GdVO_{4} crystal produces a transition into the vertical TEM_{01} mode, due to the stronger thermal lens in the sagittal plane. The accuracy with which the cylindrical lenses have to be positioned to satisfy the condition of Eq. (4) we determined to be about ±0.1mm.

To verify that the temperature distribution in the Nd:GdVO_{4} gain medium does indeed satisfy Eq. (4), we analyzed the propagation characteristics of the laser beam emitted from the 67cm cavity. Figure 5 presents the results of using the knife-edge technique [24] to determine the *M*
^{2} beam quality and embedded TEM_{00} focal mode size *w*
_{0} at the 8% output coupler in both the tangential (*x*) and sagittal (*y*) directions. The data for *w*
_{0} as a function of diode drive current (Fig. 5(a)) clearly show that the fundamental mode is indeed circular to within the roughly ±5% error of the measurement technique. The trend towards a smaller spot size at the output coupler as the strength of the thermal lens increases is expected for a symmetric laser resonator. The smallest spot size of ~350µm at 38A corresponds well to the confocal cavity condition where *w*
_{0}=337µm is expected, and it represents the point beyond which the laser output becomes distinctly more multimode and unstable.

The values of ${M}_{x}^{2}$
and ${M}_{y}^{2}$
shown in Fig. 5(b) indicate that the laser produces a high-quality TEM_{00}-mode output $\left({M}^{2}=\sqrt{{M}_{x}^{2}{M}_{y}^{2}}<1.1\right)$ up to a drive current of ~25A, where the output power is 6.5W. Between 25 and 29A, the value of ${M}_{x}^{2}$
increases to ~1.8 while the beam quality in the sagittal plane remains very high. In this region, the increase in ${M}_{x}^{2}$
is due to some additional mode structure in the tangential plane outside the strong central circular TEM_{00} mode. This behavior is expected since (i) the ~1×5mm^{2} pumped region in the gain medium (Fig. 3) provides gain for low-power higher-order horizontal mode oscillation outside the ~2mm TEM_{00} mode width; and (ii) there are no mode-control apertures in the cavity. At higher drive currents, aberrations in the sagittal pump-induced thermal lens [25] and the smaller TEM_{00} mode size in the gain medium (<500µm) allow higher-order vertical mode oscillation, causing the observed increase in ${M}_{y}^{2}$
. Nonetheless, throughout the first stability region for this 67cm cavity length (i.e., up to the confocal edge at ~38A), the diodepumped Nd:GdVO_{4} laser produces a low-divergence cw output with an overall *M*
^{2} beam quality factor of less than 2. As the flat-flat cavity length is increased, the laser generates a higher spatial beam quality, but at a reduced output power (Fig. 4), owing to the larger TEM_{00} mode size that needs to be supported at longer thermal focal lengths of the gain medium (i.e., lower pump powers). On the other hand, as expected, the higher-power shorter resonators support higher-order spatial mode beams, although the measured *M*
^{2} is only ~5 even for the 20W output power obtained for the 32cm cavity length.

The embedded TEM_{00} spot sizes *w*
_{0} measured at the output coupler (Fig. 5(a)) allow the power of the thermal lens in the diode-pumped Nd:GdVO_{4} gain medium to be extracted as a function of the diode drive current (or pump power). The results of this calculation are shown in Fig. 6. As expected, the thermal focal power (1/*f*
_{T}) is a linear function of the pump power (i.e., LD drive current for the presented range), and zero focal power is consistent with the ~14A LD threshold current. This set of data is used to define the boundary (confocal condition) between the first and second stability regions for the symmetric resonators in Fig. 4; that is, for each cavity length *L*, the drive current corresponding to the condition *f*_{T}
=*L*/2 is found to allow the boundary (solid black line) to be plotted.

It is interesting to note that thermal focal powers for diode-pumped Nd:GdVO_{4} are expected to be significantly less than those for either Nd:YAG or Nd:YVO_{4} under similar *αl*≈2 absorption conditions. Indeed, a preliminary experiment with a 3×5×10mm 0.9at.%-doped Nd:YAG crystal indicated the presence of a thermal lens with about twice the focusing power, despite the similar thermal conductivity of YAG and GdVO_{4} (κ=11-12Wm^{-1}K^{-1}) [2,3,22]. This is in agreement with both an approximately 50% larger value of $\frac{\mathit{dn}}{\mathit{dT}}$ and a substantially (~5x) greater thermal expansion coefficient α
_{T}
[2]. For Nd:YVO_{4}, the thermal lens is expected to be even stronger, mainly due to the significantly lower thermal conductivity of ~5Wm^{-1}K^{-1} [3].

## 5. Summary

We have developed an efficient diode-pumped head for Brewster-cut solid-state laser crystals in which the aspect ratio of the elliptical pumped region can be manipulated to shape a radially symmetric (or spherical) thermal lens for a laser resonator, despite the inherent astigmatism of the Brewster crystal geometry. The near-longitudinal double-end-pumped design also removes the requirement for dichroic and AR dielectric coatings while exploiting the intrinsic elliptical output from 1cm-wide pump diode arrays to provide good pump-laser spatial mode matching in the gain medium. The effectiveness of this diode pumping scheme is demonstrated using Nd:GdVO_{4} – a laser crystal for which it appears to be well suited. The Nd:GdVO_{4} laser delivers over 20W of output power at 1063nm when pumped with more than 49W of 808nm radiation from the two utilized 40W-rated diode arrays. The laser has an initial optical-to-optical slope efficiency of 37(±2)%, which compares tolerably well with many collinear end-pumped geometries [3,5,19].

Analysis of the spatial mode characteristics of the output from a 67cm-long symmetric laser resonator (with the gain medium at the center) verified that the pump-induced thermal lens in the Brewster-cut Nd:GdVO_{4} crystal was indeed spherical to the flat-flat cavity. Moreover, the theoretical pumping arrangement required to meet this condition (defined by Eq. (4)) is consistent with the results of an exact ray tracing for the pump radiation emitted by the LDs. The beam analysis also showed that laser resonators incorporating this diode-pumped head design will be capable of generating outputs with a high spatial beam quality – the 67cm-long Nd:GdVO_{4} cavity produced stable laser output with *M*
^{2}<2 throughout stability region I without the use of an intracavity aperture.

The laser head design is also sufficiently flexible to incorporate other Brewster-cut solid-state gain media for which the realization of a stigmatic spherical thermal lens requires a different aspect ratio for the elliptical pumped region (Eq. (4)). We note that for Nd:YAG, little or no stress-induced depolarization [2] is expected because the heat conduction is orthogonal to the Brewster plane and hence to the laser polarization. Further, we note that the head design would, in principle, also allow the use of fiber-coupled LD pump sources, although probably at the expense of increased initial construction and operating costs. In such a case, only an appropriate set of pump optics would be required to shape the initially circular fiber output into the required elliptical aspect ratio in the gain medium.

## Acknowledgments

The authors gratefully acknowledge the expert assistance of Kevin Lynch and the Machine Shop of the Physics Department at the University of Illinois at Chicago.

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