## Abstract

A parameter *x* is introduced to characterize the strength of thermal lens spherical aberration, whose influences on resonator’s stable zones are analyzed theoretically. Some new and helpful results are obtained. For symmetrical plane-plane cavity, spherical aberration has just influence on the back edge of stable zone. For asymmetrical plane-plane cavity, spherical aberration has influence on the back edges of the two stable zones and the front edge of the second stable zone. Effects of transverse mode collapsing to TEM_{00} mode and stable zones separation of different order’s transverse modes are pointed out, which is the foundation of TEM_{00} mode output power scaling for solid state laser oscillator. Influences of parameters such as resonator’s long arm length, short arm length, and pump beam radius on the extent to which of stable zones separation of different order transverse modes are discussed. An experimental setup of a high power diodes dual-end pumped Nd:YVO_{4} TEM_{00} mode laser oscillator is built up and investigated experimentally. 51.2 W TEM_{00} mode output power in CW operation is achieved with an optical-to-optical efficiency of about 50% and beam quality factor M^{2} being 1.2.

© 2012 OSA

## 1. Introduction

Power scaling for high power TEM_{00} mode laser output from end pumped solid state laser is an active domain in laser technology [1–4]. Due to the easily mode matching between the pump beam and the volume of the fundamental transverse mode, end pumped configuration is an effective scheme to obtain TEM_{00} mode laser output with high output power and high efficiency [5]. However, when laser crystal is high power pumped, especially end pumped, the thermal lens is no longer a perfect lens whose refractive index distribution has a parabolic profile, but one with spherical aberration accompanied [6], which is an obstacle for TEM_{00} mode power scaling.

It is now well known that spherical aberration is originated from the non-uniform temperature distribution in strong pumped crystal, which in turn originated from non-uniform pump profile, dependence of thermal conductivity *K* and thermo-optical coefficient *dn/dT* on temperature distribution [6]. Generally, spherical aberration is regarded as a major obstacle in power scaling for high power TEM_{00} mode laser [7]. When the laser crystal is high power pumped, especially for end pumped laser, spherical aberration of thermal lens becomes more and more deleterious, which results in notable diffraction loss for TEM_{00} mode laser beam and degrade the beam quality by coupling TEM_{00} mode laser to higher modes. N. Hodgson et al. applied the Fox-Li algorithm to analyze the influence of spherical aberration on stable resonators. The numerical results showed that the diffraction losses increase for stable resonators operated near the limit of stability [6]. By analyzing laser resonators with aberrations presented, I. Buske et al. pointed that it is not possible to obtain high output power and good beam quality simultaneously in the presence of severe aberrations [8].

For compensating for the spherical aberration, some methods are brought forward and realized in experiments. E. Leibush et al. developed a two dimensional Monte-Carlo based simulation package to predict and adjust the pumping distribution in crystal for elimination of spherical aberration at needed pump power [9]. Y. Lumer et al. uses spherical aberration corrector composed of a pair of spherical lenses with short focal length outside the cavity and a specially designed intra-cavity phase plate purchased from Asphericon to compensate for spherical aberration, allowing the resonator performance to be significantly improved [10]. Another more active method is adopting adaptive optics correction system, which has been shown in a Nd:YVO_{4} master oscillator and two Nd:YAG power amplifiers by using a micro-machined deformable mirror to correct spherical aberrations and thus increase output power and beam quality by U. Wittrock [11].

In addition to compensation of spherical aberration, people can also take a proactive approach to avoid the influence of spherical aberration. Spherical aberration scales quartically with the ratio of the laser mode radius to the pump light radius (i.e. filling factor) and larger filling factor increases the diffraction loss. So people can minimize the influence of spherical aberration by reducing filling factor. However, it does not mean that the filling factor should be as small as possible, because smaller filling factor decreases the extraction efficiency, and hence the output power. Therefore, people should make a tradeoff and there exists an optimum value for filling factor. Recently, C. Liu et al. reported that the spherical aberration effect determines the optimum value of beam filling factor which plays an important role in the design of a laser system [12,13]. They recommend that the optimum value of the beam filling factor should be determined by the experimental results according to the spherical aberration effect. With a beam filling factor of 0.61, they extracted 61 W TEM_{00} mode laser output from a flash lamp pumped birefringence compensated two-rod Nd:YAG laser in CW operation, which is the state of the art for a lamp pumped laser [14].

All the above mentioned are negative aspects of spherical aberration, because people need to compensate or avoid them. But in fact, there is a positive aspect really existed for spherical aberration. Spherical aberration can also be utilized to improve the brightness of laser oscillator, which is based on the fact that spherical aberration manifests itself by making dioptric power *D* of thermal lens become a radial dependent function *D(r)* instead of a constant *D _{0}*. C. Kennedy pointed out that beam quality can be degraded or improved, depending on resonator g-factor, however, efficient power extraction is problematic [15]. A. M. Bonnefois obtained 215 W CW output power with a M

^{2}factor of M

^{2}= 1.3/1.5 using the spherical aberration of laser rods. Their optical to optical efficiency is 10.4% [16]. I. Buske et al. concluded that aberrations only have a significant influence if the resonator is operated near the geometrical stability limits, but they did not explain what the influence are indeed [8]. In addition, some authors has pointed out that it is possible to operate in a region of the stability curve where only the fundamental mode is stable and higher orders are unstable, which is just qualitative. To the best of our knowledge, there has not been a detailed quantitative investigation of spherical aberration’s influence on resonator’s stable zones and how each parameter affects the spherical aberration based TEM

_{00}mode resonator design.

In this paper, by introducing a new parameter *x* that characterizes the strength of spherical aberration, we give some new insights on spherical aberration’s influence on resonator’s stable zones and make a comparison with the results obtained previously. The influences of several key parameters involved in resonator on the cavity design are analyzed theoretically. Some guidelines for optimizing the spherical aberration based TEM_{00} mode resonator are given and the way to higher TEM_{00} mode laser output is pointed out. Finally, a high power diodes double ends pumped Nd:YVO_{4} laser oscillator setup is built up and investigated experimentally. The experimental results verify and support the prediction of our theoretical analysis.

## 2. Theory and discussions

#### 2.1 Basic theory of thermal lens with spherical aberration in a strongly pumped laser rod

In solid-state lasers, the pump-induced residual heat in gain medium always causes that the temperature varies over the cross section of the gain medium because of the finite thermal conductivity and the non-uniformity of the pump radiation. This change in the temperature results in a change in the refractive index of the medium and makes the gain medium act as a lens. This is known as the thermal lens effect [17,18]. The optical path difference (OPD) function of an ideal thermal lens is parabolic. So the thermal lens has a constant focal length of *f _{0}*. However, thermal loading in actual gain medium always introduces spherical aberration of the thermal lens, especially in strongly pumped laser rods [6]. The wording “spherical aberration” is widely used in geometrical optics to describe the fact that rays which are parallel to the optical axis are focused at different points, depending on the distance to the optical axis. Here, in laser optics, many researchers also use this wording, “spherical aberration”, to describe the deviation from parabolic shape of the OPD and the radial variation of focal length of the thermal lens. That is, the focal length of thermal lens with spherical aberration is a radial dependent quadratic function

*f*(

*r*) instead of a constant

*f*.

_{0}In this paper, the attention will be focused on how the focal length of thermal lens influences resonator’s stable zones. We use an “effective” focal length *f _{T}*(

*ω*) with different mode radii [19,20]. We characterize the focal length using a quadratic Eq. (1) for a given mode radius, which is from Ref [19]. and has been modified:

_{L}*f*(4

_{0}= ω_{p}^{2}/*A*) is thermal lens’s focal length for paraxial zone,

_{0}*ω*is the oscillating laser mode radius,

_{L}*ω*is the average pump beam radius, and

_{P}*x*is the parameter we introduced to characterize the strength of spherical aberration.

*A*is defined as

_{0}*A*(

_{0}= η_{h}P_{abs}*dn/dT*)

*/*4

*πK*, where

*η*is thermal load coefficient,

_{h}*P*is absorbed pump power,

_{abs}*dn/dT*is thermal-optical coefficient, and

*K*is thermal conductivity. Because we use a dual-end composite Nd:YVO

_{4}crystal in experiments, end-faces bulging out and the stress-induced birefringence effect are neglected in

*A*.

_{0}The parameter *x* in Eq. (1) represents the strength of spherical aberration. If *x* = 0 is applied in Eq. (1), the focal length *f _{T}* goes to a constant

*f*, which means an ideal thermal lens without spherical aberration. For an actual thermal lens with spherical aberration, one could always have

_{0}*x*> 0 in Eq. (1). That is, the thermal lens shows longer focal length for a laser beam with larger diameter.

#### 2.2 Influence of spherical aberration on symmetrical resonator

In this paragraph, we will focus our attention on the influence of spherical aberration on symmetrical resonator and on asymmetrical resonator in next paragraph. Spherical aberration makes beams with different mode radii see different thermal lens’s focal length, as a result of which the beams at different radial positions have different stable zones characterization.

When round cavity mirrors employed, the eigenmodes supported by resonator can be expressed as Laguerre-Gaussian function. Then, the beam radius can be written as [21]:

where,*p*and

*l*are transverse mode’s orders and

*ω*is fundamental transverse mode beam radius. So, the beam radius for the first three lowest order modes are

_{0}*ω*,

_{00}= ω_{0}*ω*, and

_{01}= 1.414ω_{0}*ω*. When spherical aberration is not considered, the three modes see same thermal lens’s focal length and have same stable zones, which are shown in Fig. 1(a) exampled with a plane-plane cavity and two arm’s length being 310 mm. When spherical aberration is considered in Eq. (1), the three modes see different thermal lens’s focal length and have different stable zones. Figure 1(b) shows the simulation results when a simple suppose that

_{10}= 1.732ω_{0}*f*,

_{01}= 1.1 × f_{00}*f*is employed. We show it here for comparing with our new results given in Fig. 2 . However, we should notice a fact that the focal length seen by TEM

_{10}= 1.2 × f_{00}_{01}mode and TEM

_{10}mode are not always 1.1 or 1.2 times of the focal length seen by TEM

_{00}mode under different pump powers, because of the fact that the mode radius are changing with the pump power. The focal length of thermal lens can affect the beam radius, and in turn, different beam radius can see different focal length of thermal lens, meaning that they two are interrelated instead of isolated.

For stable resonators, beam radius in crystal can be expressed as [22]:

*g*,

_{1}* = 1-Dd_{2}, g_{2}* = 1-Dd_{1}, L* = d_{1}+ d_{2}-Dd_{1}d_{2}*λ*is laser wavelength,

*D*is the dioptric power and

*d*(i = 1,2) is the resonator’s arm length.

_{i}Obviously, focal length of thermal lens influences the beam radius *ω _{L}* in crystal by Eq. (3) and beam radius

*ω*in crystal in turn can influences the focal length of thermal lens seen by itself by Eq. (1), which is a iterative and coupling process. In the limit condition that spherical aberration coefficient

_{L}*x*in Eq. (1) goes to zero, the interrelation between focal length of thermal lens and beam radius

*ω*in crystal disappears. When spherical aberration coefficient

_{L}*x*is nonzero, we can solve the simultaneous Eq. (1) and Eq. (3) with different

*x*.

Figure 2 shows the numerical calculation results of spherical aberration’s influence on plane-plane symmetrical cavity’s stable zones for TEM_{00}, TEM_{01} and TEM_{10} modes when Eq. (1) and Eq. (3) are simultaneously solved with spherical aberration coefficient *x* = 0.1, *x* = 0.2, *x* = 0.3 and *x* = 0.5. As a reference, the TEM_{00} mode stable zone obtained without spherical aberration considered is also shown in Fig. 2. Compared with that shown in Fig. 1, some new results appear in Fig. 2. Spherical aberration affects not only the extent of high order mode’s stable zone, but also that of fundamental mode’s. The strength of spherical aberration directly determines the extent to which the stable zone’s back edge extends. Regardless of higher order modes and fundamental mode, the beam radius in crystal reaches their maximum value at almost the same pump power, which is higher for condition with larger spherical aberration *x*. In sum, for plane-plane symmetrical resonator, the introduction of spherical aberration has main influence on back edge of stable zone, the extent to which is determined by the strength of spherical aberration.

#### 2.3 Influence of spherical aberration on asymmetrical resonator

A plane-plane asymmetrical resonator forms when the length of two arms are different. By adequate choice of the resonators asymmetry of output coupler and HR mirror arm-lengths two stability ranges appears. With strong asymmetry, a variation of the shorter length can be used to shift the stability range with respect to thermal lens power, and variation of the longer one mainly alters the mode diameter inside the laser crystals and consequently the width of stability range, which gives full control of the stability ranges of the system. When spherical aberration is introduced, the radially varying dioptric power of thermal lens makes larger beam see a longer focal length, which can shift the stability zone of the resonator for higher-order modes to higher pump powers.

Figure 3(a)
shows the stable zones diagrams for plane-plane asymmetrical cavity without spherical aberration considered, calculated by solving Eq. (3) with a long arm length of d_{2} = 310 mm, short arm length of d_{1} = 55 mm, pump beam radius of ω_{p} = 0.8 mm. Stable zones for transverse modes of different orders have the same ranges. Figure 3(b) shows the stable zones diagrams for plane-plane asymmetrical cavity with spherical aberration considered when a simple suppose that *f _{01} = 1.1 × f_{00}*,

*f*is employed. Some new phenomenon appears. Firstly, for the first stable zone (corresponding to low pump power), the presence of spherical aberration broadens its back edge. Secondly, for the second stable zone (corresponding to high pump power), transverse modes with different order experience different focal length, which makes each transverse mode has its own critical pump power from which the second stable zone enters. Therefore, separation of stable zones is formed because of the aspherical profile of the thermal lens, which means a large discrimination between fundamental mode and higher-order modes. Efficient TEM

_{10}= 1.2 × f_{00}_{00}mode operation can be achieved if one chooses the beginning of the second stable zone of the TEM

_{00}mode as the working point for the TEM

_{00}mode without an internal aperture needed. When combined with proper adjustment of two arm’s lengths, large TEM

_{00}mode volume and appropriate working point can be realized.

However, one should note that not all plane-plane asymmetrical resonators have such efficient separation effect of stable zones, at least not so efficient. By analyzing the influence of each parameter involved in the separation effect of stable zones, we summarize some guidelines that support efficient separation effect of stable zones. (1). The length of short arm should be short enough. In this way, the second stable zone can be shifted to higher pump power and efficient separation of stable zones can be achieved. Figure 4(a)
and 4(b) shows this trend. Keeping a long arm length of d_{2} = 310 mm and pump beam radius of ω_{p} = 0.8 mm, when short arm length is increased to 550 mm, the separation effect of stable zones strengthens. When long arm length is decreased to 110 mm, the separation effect of stable zones weakens. (2). The length of long arm should be long enough. In this way, △D narrows according to the relationship of ω_{30}^{2}△D = 2λ/π and efficient separation of stable zones can be achieved. However, it is at the cost of increasing the beam radius ω_{30} in crystal. This should be carefully considered based on the reality. Figure 4(c) and 4(d) shows this trend. Keeping a short arm length of d_{1} = 55 mm and pump beam radius of ω_{p} = 0.8 mm, when long arm length is increased to 550 mm, the separation effect of stable zones strengthens. When short arm length is decreased to 110 mm, the separation effect of stable zones weakens. (3). The pump beam radius should be selected carefully. If the pump beam radius increases and pump power keeps same, the focal length of paraxial zone certainly will increases. Under the condition that keeping the resonator’s configuration unchanged, higher pump power is needed to shift the second stable zone. Figure 4(e) and 4(f) shows this trend. Appropriate pump beam radius should be selected by considering the available pump power, available configuration room and laser crystal’s thermal characteristics, comprehensively.

Similar to Fig. 2, Fig. 5
shows the numerical calculation results of influences of spherical aberration on plane-plane asymmetrical cavity’s stable zones for TEM_{00}, TEM_{01} and TEM_{10} mode when Eq. (1) and Eq. (3) are simultaneously solved with spherical aberration coefficient *x* = 0.1, *x* = 0.2, *x* = 0.3 and *x* = 0.5. The TEM_{00} mode stable zone obtained without spherical aberration considered is also shown in Fig. 5 as a reference. Compared with that shown in Fig. 3, some new results appear in Fig. 5. For the first stable zone, spherical aberration mainly affects back edge of stable zone and the strength of spherical aberration directly determines the extent to which the stable zone’s back edge extends. Regardless of higher order modes and fundamental mode, the beam radius in crystal reaches their maximum value at almost the same pump power, which is higher for condition with larger spherical aberration *x*. For second stable zone, spherical aberration makes the critical pump power at which the second stable zone enters different for transverse modes of different orders, which forms the separation effect of sable zones of transverse modes. Larger spherical aberration coefficient *x* leads to larger separation effect. But the back edges of transverse modes with different orders converge at almost same pump power, which is similar to that of the first stable zone.

In addition, three interesting phenomena can be predicted by what shown in Fig. 5. First, the presence of spherical aberration makes the unstable zone between the two stable zones narrows and even disappears if the spherical aberration is strong enough. In experiments, it can leads to that the output power drops in unstable zone but cannot drops to zero except that the diffraction loss is high enough to suppress the possible multimode laser oscillation. Second, transverse mode collapsed to near TEM_{00} mode can be predicted. The shapes of stable zone change from U-shape to √-shape and the short side of “√” shortens with the spherical aberration coefficient *x* increased, which means that the beam radius supported by the resonator will have a sudden break from a large beam radius in so-called unstable zone to a small beam radius when the second stable zone is reached. Third, it predicts larger TEM_{00} mode working zone with spherical aberration coefficient *x* increased, shown by the shadow region in Fig. 5.

## 3. Experiments

The schematic diagram of the experimental setup is shown in Fig. 6
, where a simple plane-plane cavity with a diodes-double-end pumped configuration has been adopted. In a high power diode pumped solid state laser system, laser gain mediums with low doped levels will have better performance than that with high doped levels, because the lower absorption coefficient of a low doped level laser crystal can uniform the distribution of the pumping beam, and mitigate the thermal problem, which is very important for a high-power diode-pumped solid-state laser. Moreover, it can also tolerate much higher incident pump power. Thus, a 3 × 3 × (2 + 16 + 2) mm^{3} Nd:YVO_{4} composite crystal with a Nd^{3+}-doped level as low as 0.3 atm.% was used as the laser gain medium. Two high-brightness and high-power fiber-coupler laser diode (made by DILAS Inc.) with each maximum output power of 50 W and wavelength of 808 nm are used as the double ends pumping source. The output fibers have a fiber-core diameter of 400 μm and a numerical aperture of 0.22. For each one, the fiber end is imaged to the closer end face of Nd:YVO_{4} laser crystal with a diameter of 800 μm through a pair of positive aspherical lenses. For better thermal contact and reducing the deleterious thermal effects, the Nd:YVO_{4} crystal is wrapped with a piece of 0.1 mm thick indium foil and mounted in a four sides edge water-cooled copper. The temperature of cooler for crystal is set at 18 degrees Celsius and 22 degrees Celsius for the laser diode’s cooler. The both end faces of the Nd:YVO_{4} crystal are high transmittance (HT) coated at 808 nm and 1064 nm. Two dichotic mirrors, high-reflectivity (HR) coated at 1064 nm for 45° incidence angle and HT coated at 808 nm, are used to form a “U”-type resonator with a mirror HR coated at 1064 nm and an output coupler with 50% transmittance for 1064 nm. For efficient TEM_{00} mode operation, the lengths of resonator’s two arms are optimized to be 55 mm and 310 mm.

Figure 7
shows the CW output power characteristics of the dual-end pumped Nd:YVO_{4} laser with the optimized asymmetrical cavity configuration and a 50% OC transmission. The blue triangle figures the output power with optimized asymmetric cavity and a maximum TEM_{00} mode power of 51.2 W is achieved at pump power of 104 W. The red shadow zone represents the TEM_{00} mode operation zone. Figure 8
shows the two- and three-dimensional beam profiles captured at TEM_{00} mode output power of 51.2 W, which are taken at 2 m far away from the laser exit. Its beam quality factor M^{2} is measured to be 1.2, shown in Fig. 9
.

In experiments, we can observe that the output power drops in unstable zone but not to zero, just as shown in Fig. 7, which can be attributed to the existence of spherical aberration. When the pump power is gradually increased to 98 W, the laser mode will suddenly collapse to TEM_{00} mode without any halo light’s concomitance. This TEM_{00} mode operation can keep up to pump power of 108 W, after which the laser will oscillate in multimode again and output power decreases. This process is shown in Fig. 10
. In experiments, we also pump the Nd:YVO_{4} crystal with just one laser diode. Though it can deliver about 25 W TEM_{00} mode laser output power, its TEM_{00} mode operation zone is very narrow compared with the dual-end pumped configuration.

The TEM_{00} mode output power is finally restricted to 54 W in our experiments, obtained with a very short short-arm-length and laser diode’s full load operation. So it is pump power limited. Higher TEM_{00} output power can be expected with stronger pump source. However, that is not all. We think there are some principles that should be considered. Firstly, if we want to further scale the TEM_{00} mode output power with higher pump power, the length of short arm should be further shortened, which will lead to the beam radius on HR mirror too small and increase the damage possibility. In this condition, one can consider making the short arm’s length longer and beam radius on HR mirror larger by using an appropriate concave HR mirror. Secondly, we need choose an appropriate pump beam radius according to the available pump power and the crystal’s thermal characteristics. For keeping the crystal’s thermal lens dioptric power same, higher pump power available will inevitably lead to larger pump beam radius. Third, we should choose a proper laser crystal with enough large cross section to support the pump beam radius. In addition, larger laser beam radius in crystal necessitates a longer length of long arm. Longer length of long arm will make the laser configuration not so compact and the pulse width longer when the laser is Q-switched. However, they can be easily settled by inserting an appropriate positive lens to shorten the length of long arm or incorporating a cavity damping technique. Anyway, higher TEM_{00} mode output power can be expected with our method, combining some other artifice when needed.

## 4. Conclusions

By introducing a new parameter, spherical aberration coefficient *x*, we give a thoroughly comprehensive analysis on the spherical aberration’s influence on the stable zones of plane-plane resonators. Theoretically, it is very convenient for us to include the parameter x in the calculation process of the stable zones for modes with different orders. Some new results different to that what have been reported before is obtained, including spherical aberration’s influence on the front and back edges of stable zones, the shapes of stable zone change from U-shape to √-shape, prediction of the narrow of unstable zone between the two stable zones, prediction of that the output power drops in unstable zone but cannot drops to zero, prediction of that transverse mode collapsed to near TEM_{00} mode, and prediction of that larger TEM_{00} mode working zone can be achieved with larger spherical aberration coefficient *x*. All these new results are observed in our experiment, which is performed with a homemade high power dual-end pumped Nd:YVO_{4} laser oscillator. Guidelines for scaling TEM_{00} mode output power directly from an oscillator are analyzed and discussed.

## Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant No. 60908013).

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