New geometries for solid state lasers like the thin disc, different advanced thin slab designs or the fiber laser more and more oust the classic rod laser in the cw operation regime. This is mostly because of their advantages in the handling of the thermal load. However, in the domain of ns-pulses with higher pulse energies neither geometries with a small cross section like the fiber nor geometries inherently supplying only low or moderate gain factors like the thin disc design are suitable. To realize larger pulse energies for ns-pulses a sufficient cross section of the beam and hence the laser active material is needed to avoid damage. Furthermore a suitable gain factor for an easy saturation of the gain is desirable. Thus, bulk geometries are the appropriate choice in this case. This could be either a rod or slab-type geometry. The isotropic rod suffers from thermally induced birefringence but supplies a circular symmetric beam, whereas in the slab geometry the thermally induced birefringence does not have a negative impact on the beam quality but an astigmatic beam is produced.
Therefore, rod lasers are still among the best possible geometry choices for the realization of a reasonably simple high energy pulsed laser system in the Joule range. In this paper we consider the consequences of the thermal load arising at higher average pump powers in Nd:YAG rod lasers with special regard to the recently available core doped ceramics rods [1
, BP501, F-74339 La Balme de Sillingy cedex, France
]. Laser rods made of ceramics essentially show equal thermo-mechanical properties as crystalline rods [2
High efficient 2% Nd:yttrium aluminum garnet ceramic laser
Appl. Phys. Lett.
Potential of Ceramic YAG Lasers
]. The core doped ceramics rods are laser active in the Nd-doped core, only. The cladding consists of undoped YAG ceramics. This core doped rod geometry is interesting for various reasons (see e.g. [4
Diode End-Pumped Core-Doped Ceramic Nd:YAG Laser
]). In this paper we are concentrating on the potential advantage that in side pumped arrangements the core doped rod can accommodate wider Gaussian profiles in the laser active region without truncating its wings. This will lead to a more efficient use of the built-up inversion in rod amplifiers without too a severe impact of diffraction at the rod’s aperture.
Many consequences of the thermal load in solid state laser rods are known as isolated implications. But hardly anywhere in the literature a report on the quantitative consequences of the arising aberrations of the refractive index profile in the laser rod on the beam quality can be found. We specifically consider the side pumped side cooled laser design. The different impacts on the beam quality which are higher order aberrations of the refractive index profile in the laser rod and the diffraction at the boundary aperture of the laser rod are compared quantitatively with special regard to the M2
-parameter. The investigation is performed on the basis of a finite elements analysis (FEA) model implemented in Femlab [5
Femlab 3.1, Comsol, 1994-2004
]. Thermally induced birefringence is neglected in our considerations since it can be well compensated for [6
A novel approach for compensation of birefringence in cylindrical Nd:YAG rods
Opt. Quantum Electron.
Quasi-continuous-wave birefringence-compensated single- and double-rod Nd:YAG lasers
]. Also at this stage of our investigation no modification of the beam profiles by the gain in the laser rods is included either.
In the second section of this paper our FEA model is described. A circular symmetric two dimensional model is used to calculate the temperature distributions for the rod cross section for different pump scenarios. From the temperature profiles subsequently the refractive index distributions in the laser rods can be calculated. Although we consider pulsed pumped laser materials the thermal load is considered as temporally constant load. This is justified in good approximation for higher repetition rates of the pump pulses above 100 Hz. In the third section aspects of the thermal load of the classic rod design are investigated. The resulting refractive index profiles for differently shaped pump distributions are presented. A diffraction limited beam is then propagated through the laser rods with the different refractive index profiles. The resulting M2-values are evaluated numerically and compared for the different pump scenarios. In section 4 these considerations of the impact of the thermal load on the beam quality of a beam passing through the amplifier are extended to core doped laser rods.
2. Finite elements model to calculate refractive index profiles in laser rods
We use a finite elements model to calculate the temperature distribution of the laser rod cross section in side pumped side cooled designs. The model we use is a stationary two dimensional model. It assumes that there is a circular symmetric pump distribution that stays constant along the laser rod axis. The rod is assumed to be homogeneously cooled on its outer surface (see Fig. 1
The model is based on the partial differential equations (PDE) for the stationary heat convection and conduction. To solve the PDE’s for laser rods with specific boundary conditions the software Femlab from Comsol [5
Femlab 3.1, Comsol, 1994-2004
] was utilized. The used material parameters for the Nd:YAG rod and the thin film coefficient of the water cooling flow are listed in Table 1
Table 1. Nd:YAG-Parameters used in FEA-Modeling
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Fig. 1. Principal sketch for FEA-model.
For all calculations throughout this paper a heat generation power of 91 W was assumed. The fractional thermal load was calculated solely by the quantum defect, 1 - 808 nm/1064 nm which means that bypass relaxations are neglected. At this fractional thermal load of 0.241 it follows that the calculations are performed for an absorbed pump power of 378 W.
In most cases, when designing the laser heads a top hat like distribution of the incident pump light across the laser rod is desired. To investigate the impact of the shape of the radial pump distribution we investigated 4 different cases (see Fig. 2
): a radially constant or top hat distribution (case 1), two parabolic distributions (cases 2 & 3), and a hybrid distribution typical for many real pump chambers (case 4). Starting from these different excitation profiles as input data, the FEA-model yields the radial temperature distribution across the laser rod. The resulting refractive index profiles are calculated from the temperature distribution by n(T) = n0
+ 0.0548 K-2*
T - 0.00005 K-3*
]. Described by the photo-elastic effect the refractive index is also dependent on the thermally induced stress. However in Nd:YAG this effect is about one order of magnitude smaller compared to the direct dependence of the refractive index on the temperature [11
Solid-State Laser Engineering
), Chapter 7.1
]. Thus the stress induced change of the refractive index is neglected for our considerations.
Fig. 2. Pump profiles for an absorbed pump power of 378W
When a beam passes through a material with a radially dependent refractive index profile the beam quality usually deteriorates. However, in the special case of a parabolic refractive index profile the beam quality of the transmitted beam remains constant within the paraxial description of the propagation.
If the heat conductivity in Nd:YAG was not temperature dependent, a homogeneous pump distribution in the laser rod and hence homogeneous heat generation would result in a purely parabolic temperature distribution. Assuming in addition that dn/dT of Nd:YAG was not temperature dependent, the parabolic temperature distribution would lead to a parabolic refractive index distribution. Since the heat conductivity in Nd:YAG is temperature dependent and also the heat generation in the laser rods usually has some deviations from the homogeneous case more or less severe deviations from an ideal parabolic refractive index profile result. To quantify the strength of the aberrations of the profile we fit 4th order polynomial functions to the refractive index profiles:
The pure parabolic refractive index profiles generate the lensing effect, without disturbing the beam quality. Therefore the potential deterioration of the beam quality by the refractive index profile can be quantified by the 4th-order aberration coefficient C4 of the refractive index profile. The focal length f of the thermal lens is calculated from a parabolic fit of the resulting refractive index profile:
3. Results of FEA model for classic laser rod
First the FEA model is applied to a 9 mm diameter laser rod. As expected the best result regarding the size of the 4th
order coefficient C4
is achieved by the top hat like excitation distribution (see case 1 in Fig. 3
and Table 2
). The 4th
order aberration becomes stronger for the parabolic heat generation distributions (cases 2 and 3). The strength of the 4th
-order aberration of the hybrid distribution of case 4 ranges in between the two parabolic cases.
Fig. 3. Refractive index profiles for 378 W of absorbed pump power for Nd:YAG-rod with radius rrod = 4.5 mm.
Table 2. Coefficients from equation 1, resulting focal length f, and temperature at the rod’s boundary and center for Nd:YAG rod with radius of rrod = 4.5 mm and length Lrod = 12 cm corresponding to the refractive index profiles shown in Fig. 3 for 378 W of absorbed pump power.
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The effect of these resulting refractive index profiles on the beam quality of a transmitted beam are investigated by numerically propagating a beam through them. We assume the rod to act as a thin phase plate with an aperture. A phase shift function is evaluated from the refractive index distribution in the laser rod:
A collimated Gaussian input beam described by the field Ein with a beam radius of 2/3 of the rod radius is propagated from the rod’s endface to different distances Li by numerically solving the Fresnel integral:
λ) denoting the Fresnel number, r1
the radial coordinate at the rod’s endface plane, r2
the coordinate in the plane at distance Li
denotes the 0th
-Bessel function. The second moment beam radius for the field distribution is evaluated at each distance Li
. These beam radii yield the caustics shown in Fig. 4
. Due to the thermal lensing effect of the rod a focus is produced. The focal positions are in agreement with the calculated focal lengths from the refractive index profiles.
Calculating the beam propagation through the rod according to the above algorithm allows considering the consequences of the 4th
order aberrations on the beam quality. It also includes diffraction effects at the rod’s aperture. To comprise the real boundary condition of the rod even better, a differential propagation using the wave equation with transparent boundary conditions [12
Opt. Fiber Technol.
] would have to be used. However, in the calculated scenarios of the propagation through the rods the Fresnel numbers are bigger then 10 and we consider collimated beams as input beams to the rods. Thus, it is justified to approximate the propagation through the rod by a “geometrically” constructed phase function followed by the propagation with the Fresnel integral for the longer distances and smaller beam radii behind the rod.
Fig. 4. Caustics of a 6 mm diameter Gaussian input beam behind the 9 mm diameter laser rod pumped with 378 W of absorbed pump power arising for the 4 different cases.
The beam radii are calculated as a second moment of the intensity since this is the accepted definition by the International Organization of Standardization (ISO) for the beam radius. Within the second moment definition the intensity gets weighted with the square of the radius (see, e.g., [13
), Chapter 2.6
]). This makes the determination of the second moment beam radius sensitive to noise contributions further away from the optical axis. This noise can be due to e.g. electronic read out noise of the CCD camera in case of an experiment or due to the discretization in case of our calculations. These facts of the case can lead to some deviation in the calculated and measured beam radii compared to the real beam radii. The effect becomes more pronounced for higher M2
-numbers and leads to even higher M2
-values if no offset is subtracted from the intensities.
However, a fit of the beam radii yields the M2
-values for the different pump distribution cases. A beam propagation factor that is yielded without considering 4th
-order aberrations can be simply calculated, too. In this case a caustic is generated from the beam radii calculated from the Fresnel integral (4) with the 4th
-order coefficient set to zero. Without 4th
-order aberrations for the 9 mm rod an M2
of 1.5 results. The comparison to the M2
of 1.5 in case 1 (see Fig. 4
) shows that in case 1 the beam quality deterioration is solely due to diffraction at the rod aperture whereas in the other cases the impact of the 4th
-order aberrations is more and more dominating the M2
The resulting M2
values correspond to the strength of the 4th
-order aberration coefficient of the refractive index profile. The impact of the aberration on the beam profiles at selected positions along the caustic is shown in Fig. 5
for the 4 different pump scenarios. The more severe the 4th
-order aberration the more distinct the modulation of the profiles appears in the right wing of the caustic. This does not only degrade the M2
-value but might also cause severe damage problems due to hot spot characteristics of the profiles.
Profiles from the caustics in Fig. 4
at different distances z. Rows 1-4 present the corresponding cases 1-4.
Depending on the specific application of the laser a certain rod diameter will be chosen. For example the realization of higher pulse energies requires bigger rod diameters. The rods considered so far had a diameter of 9 mm. Smaller pulse energies can be realized with smaller rod diameters. The ratio of cooling surface to volume to be cooled gets bigger for smaller rod diameters. This would lead to lower temperatures for smaller rod diameters. On the other hand for bigger volumes the heat density is lower at identical absolute heat generation rates. This would lead to lower average temperatures for bigger rod diameters. The average temperature in the rod is an interesting parameter because of the temperature dependence of the heat conductivity. When the resulting temperatures in the 9 mm diameter rod (see Table 2
) are compared with corresponding temperatures in a 5 mm diameter rod (see Table 3
), it can be concluded that the two above mentioned correlations almost cancel out so that the resulting temperatures in the rod center are identical within 1 K.
also shows the coefficients of the resulting refractive index profiles for the 5 mm diameter laser rod. Based on these results the theoretically expected values for a 9 mm diameter rod are calculated and presented in the lower part of the table. The absolute refractive power in the laser rods is inversely proportional to the rod’s cross section as expected. From the comparison to Table 2
it can be seen that the resulting aberrations have almost the same relative strength in both cases for the 5 mm and 9 mm rod diameter.
Table 3. Coefficients from equation 1, resulting focal length f, and temperature at the rod’s boundary and center for Nd:YAG rod with radius of rrod = 2.5 mm and length Lrod = 12 cm compared to extrapolated values for a rod with 4.5 mm radius for 378 W of absorbed pump power (compare to Table 2).
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Since the thermal conductivity in Nd:YAG is temperature dependent it is interesting to ask for the difference in the refractive power and the 4th
-order aberration for different cooling water temperatures. The results in Table 4
show that the refractive power becomes only marginally stronger when rising the cooling water temperature from 10°C to 30°C. There is a bigger fluctuation of the 4th
order coefficient of case 1 for the different cooling water temperatures. But this fluctuation appears on a low absolute level and hence is not a very significant process in its consequences on the beam quality. The relative strength of the 4th
order coefficients for the other cases stays constant within one or two percent.
Table 4. Results for Nd:YAG-rod with a radius of rrod = 2.5 mm and a length of 12 cm. The coefficients for the refractive index profiles, the focal length f and the temperatures at the rod’s boundary and center are given at three different cooling water temperatures for 378 W of absorbed pump power.
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4. Comparison to core doped rods
The upcoming of Nd:YAG ceramics allows for a new design variant of laser rods. In a two step fabrication process so called core doped rods can be manufactured. A core region of ceramic YAG is doped with Neodymium and a cladding region consists of undoped ceramic YAG. Thus, only the core is providing gain. Since the rod cross section is widened by the undoped cladding, wider Gaussian intensity distributions can be accommodated in the laser rod. Therefore the built up inversion can be used more efficiently as the average intensity in the doped part of the laser rod becomes higher. In addition the truncation of the wings of Gaussian profiles is diminished and thus the negative impact of diffraction on the beam quality should be reduced.
In this section we investigate to what end this conceptual advantage can be actually exploited taking into account the refractive index step from the doped core to the undoped cladding part. The refractive index difference is 0.005 % for a doping level of 0.8 at% [14
, BP501, F-74339 La Balme de Sillingy cedex, France, (private communication,
]. The undoped YAG has a refractive index of 1.8168 at 1064 nm. This refractive index step will cause some kind of diffraction on one hand. On the other hand the smaller refractive index of the cladding compared to the core would allow for total internal reflection up to an incident divergence of 19 mrad. But since the grain size of the Nd:YAG ceramics is of the order of 50 μm the effect will be more a scattering rather than a reflection at a toroidal surface. Also because we consider collimated input beams there will be only a minor impact of this effect on the propagating beam. Therefore we neglect the total internal reflection at the core/cladding boundary for our beam propagation calculations.
We consider two different rods. Both have a diameter of 5 mm but different core diameters of 3 mm and 4 mm to give more or less room to the wings of the Gaussian profiles. The doping level of the cores is 0.8 at%.
shows the refractive index profiles resulting from the FEA model for the cases 1-3 for the 3 mm core doped rod. In Table 5
the coefficients of the fits of theses profiles in the core area according to Eq. (1
) are given. The refractive index profile of the core doped rod and a corresponding ordinary crystalline 3 mm rod are compared. An equal amount of absorbed pump power has been assumed for the calculations. The results are presented in Table 6
. The focal lengths of the thermal lenses are very similar with somewhat smaller values in the cases 2 and 3 for the core doped rod.
Fig. 6. Refractive index profiles for the core doped Nd:YAG-rod with a radius of rrod = 2.5 mm, a doped core with rdoped = 1.5 mm, and a length of 12 cm for 378 W of absorbed pump power.
Table 5. Coefficients of the refractive index profiles of the core doped rod with rdoped = 1.5 mm core radius, rrod = 2.5 mm outer radius, and a length of 12 cm as depicted in Fig. 6 for 378 W of absorbed pump power.
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Table 6. Coefficients of the refractive index profiles of a 12 cm long crystalline Nd:YAG rod with a radius of rrod = 1.5 mm, for 378 W of absorbed pump power.
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The refractive index profiles of the 4 mm core doped rod relates to the refractive index profiles of a 4 mm crystalline Nd:YAG rod in the same manner as the 3 mm core doped rod relates to its conventional 3 mm counter part (compare Table 7
with Table 8
Fig. 7. Refractive index profiles for the core doped Nd:YAG-rod with a radius of r = 2.5 mm, a doped core with rdoped = 2 mm, and a length of 12 cm for 378 W of absorbed pump power.
Table 7. Coefficients of the refractive index profiles of the core doped rod with rdoped = 2 mm core radius rrod = 2.5 mm outer radius, and a length of 12 cm as depicted in Fig. 7 for 378 W of absorbed pump power.
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Table 8. Coefficients of the refractive index profiles of a 12 cm long crystalline Nd:YAG rod with a radius of rrod = 2 mm, for 378 W of absorbed pump power.
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Next, the impact of the calculated refractive index profiles on the beam quality of a Gaussian beam passing through the rod is investigated. The wings of the Gaussian profiles can be accommodated in the undoped part of the rod. Thereby diffraction at the rod’s aperture should be minimized. However, the resulting beam quality for equal beam radii in the core doped cases is distinctly worse compared to the conventional crystalline rods (see Table 9
). The major reason for the worse M2
-values can be deduced when the beam propagation with and without aberrations is considered. The data for three different input radii of a Gaussian beam was calculated. The comparison in Table 9
shows that the deterioration of the beam quality in case of core doped rods cannot be explained with regard to the 4th
-order aberration of the refractive index profiles. It has to be due to the refractive index step at the boundary between the doped and undoped part. This step leads to a diffraction effect similar to the aperture of the conventional rod.
As mentioned in section 3 evaluating the M2-values with high accuracy is a challenging task for bigger M2-numbers in both domains, experiment and numerical calculation. Thus, the bigger M2-values should be understood as quantitative indication for a beam quality degradation rather than an exact absolute value.
When the beam radius of a Gaussian input beam is small compared to the boundary of the doped and undoped part of the rod the M2-value of the beam is close to one. But when the beam radius gets closer to the core radius the M2-value increases significantly. The M2-value of 1.6 for the crystalline 5 mm rod is increased because of diffraction at the 2.5 mm radius aperture of the rod.
The significant deterioration of the beam quality by the refractive index step creates a big obstacle in exploiting the potential advantage of the core doped rods.
Table 9. Calculated beam qualities for a collimated Gaussian beam with different beam radii passing through the core doped Nd:YAG-rods with 1.5 mm and 2 mm core radius and 2.5 mm outer radius in comparison to a crystalline rod with 2.5 mm radius. All examples are calculated for the pump distribution case 1. M
C4 denotes the beam propagation factor for a propagation with 4th-order aberrations, M
2 is the beam quality for propagation without 4th-order aberration term C4 in the refractive index profiles shown in Fig. 6 and Fig. 7.
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