1. Introduction
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
1
.
Baikowski
Chimie
, BP501, F-74339 La Balme de Sillingy cedex, France
]. Laser rods made of ceramics essentially show equal thermo-mechanical properties as crystalline rods [
2
2
.
L.
Jianren
,
M.
Prabhu
,
X.
Jianqiu
,
K.
Ueda
,
H.
Yagi
,
T.
Yanagitani
, and
A. A.
Kaminski
, “
High efficient 2% Nd:yttrium aluminum garnet ceramic laser
,”
Appl. Phys. Lett.
78
,
3707
–
3709
(
2000
)
,
3
3
.
J.
Lu
,
M.
Prabhu
,
K
Ueda
,
H.
Yagi
,
T.
Yanagitani
,
A.
Kudryashov
, and
A. A.
Kaminski
, “
Potential of Ceramic YAG Lasers
,”
Laser Phys.
78
,
1053
–
1057
(
2001
)
]. 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
4
.
D.
Kracht
,
M.
Frede
,
D.
Freiburg
,
R.
Wilhelm
, and
C.
Fallnich
, “
Diode End-Pumped Core-Doped Ceramic Nd:YAG Laser
,”
ASSP
2005
]). 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.
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 M^{2}-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
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 |
| |
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.
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 4^{th} 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 4^{th}-order aberration coefficient C_{4} 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 4
^{th} order coefficient C
_{4} is achieved by the top hat like excitation distribution (see case 1 in
Fig. 3 and
Table 2). The 4
^{th} order aberration becomes stronger for the parabolic heat generation distributions (cases 2 and 3). The strength of the 4
^{th}-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 r_{rod} = 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 r_{rod} = 4.5 mm and length L_{rod} = 12 cm corresponding to the refractive index profiles shown in Fig. 3 for 378 W of absorbed pump power. |
| |
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 E_{in} with a beam radius of 2/3 of the rod radius is propagated from the rod’s endface to different distances L_{i} by numerically solving the Fresnel integral:
With F
_{i} = r
_{rod}/(L
_{i}λ) denoting the Fresnel number, r
_{1} the radial coordinate at the rod’s endface plane, r
_{2} the coordinate in the plane at distance L
_{i}. J
_{o} denotes the 0
^{th}-Bessel function. The second moment beam radius for the field distribution is evaluated at each distance L
_{i}. 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.
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
13
.
N.
Hodgson
and
H.
Weber
,
Optical Resonators
, (
Springer
1997
), 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 M
^{2}-numbers and leads to even higher M
^{2}-values if no offset is subtracted from the intensities.
However, a fit of the beam radii yields the M
^{2}-values for the different pump distribution cases. A beam propagation factor that is yielded without considering 4
^{th}-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 4
^{th}-order coefficient set to zero. Without 4
^{th}-order aberrations for the 9 mm rod an M
^{2} of 1.5 results. The comparison to the M
^{2} 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 4
^{th}-order aberrations is more and more dominating the M
^{2} value.
The resulting M
^{2} values correspond to the strength of the 4
^{th}-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 4
^{th}-order aberration the more distinct the modulation of the profiles appears in the right wing of the caustic. This does not only degrade the M
^{2}-value but might also cause severe damage problems due to hot spot characteristics of the profiles.
Fig. 5. 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.
Table 3 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 r_{rod} = 2.5 mm and length L_{rod} = 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). |
| |
Since the thermal conductivity in Nd:YAG is temperature dependent it is interesting to ask for the difference in the refractive power and the 4
^{th}-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 4
^{th} 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 4
^{th} order coefficients for the other cases stays constant within one or two percent.
Table 4. Results for Nd:YAG-rod with a radius of r_{rod} = 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. |
| |
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.
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%.
Fig. 6. Refractive index profiles for the core doped Nd:YAG-rod with a radius of r_{rod} = 2.5 mm, a doped core with r_{doped} = 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 r_{doped} = 1.5 mm core radius, r_{rod} = 2.5 mm outer radius, and a length of 12 cm as depicted in Fig. 6 for 378 W of absorbed pump power. |
| |
Table 6. Coefficients of the refractive index profiles of a 12 cm long crystalline Nd:YAG rod with a radius of r_{rod} = 1.5 mm, for 378 W of absorbed pump power. |
| |
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 r_{doped} = 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 r_{doped} = 2 mm core radius r_{rod} = 2.5 mm outer radius, and a length of 12 cm as depicted in Fig. 7 for 378 W of absorbed pump power. |
| |
Table 8. Coefficients of the refractive index profiles of a 12 cm long crystalline Nd:YAG rod with a radius of r_{rod} = 2 mm, for 378 W of absorbed pump power. |
| |
As mentioned in section 3 evaluating the M^{2}-values with high accuracy is a challenging task for bigger M^{2}-numbers in both domains, experiment and numerical calculation. Thus, the bigger M^{2}-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 M^{2}-value of the beam is close to one. But when the beam radius gets closer to the core radius the M^{2}-value increases significantly. The M^{2}-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
^{2}
_{C4} denotes the beam propagation factor for a propagation with 4^{th}-order aberrations, M
^{2} is the beam quality for propagation without 4^{th}-order aberration term C_{4} in the refractive index profiles shown in Fig. 6 and Fig. 7. |
| |
5. Conclusion
We have numerically investigated the thermal lens properties of core doped ceramics rods and compared them to conventional crystalline bulk Nd:YAG rods. A severe deterioration of the beam quality when passing through the core doped rods will be observed due to the refractive index jump from core to cladding. If this effect cannot be overcome the potential advantage of avoiding diffraction at a rods aperture by the core doped design is no advantage anymore. However, using core doped rods in a double pass master oscillator power amplifier design this problem could be compensated for by using a phase conjugating mirror for the back reflection toward the second amplifier pass. This would compensate for the wavefront distortion and the advantage of the relatively wider rod aperture would be maintained.
A core doped rod in a laser oscillator might create a slightly different picture. The share of the electrical field that gets distorted by the refractive index jump will experience a higher diffraction loss at the spherical mirrors. As a result the beam quality of the outcoupled eigenmode might not be that bad but in turn the losses of the resonator for the TEM_{00} might be slightly higher. On the other hand higher order modes will experience even higher losses so that the mode discrimination might be better compared to conventional crystalline rods.
But if the loss is already too high for the TEM_{00} there might be no advantage that can be exploited anymore. We started to investigate the benefits of the core doped rods in both MOPA and oscillator configurations experimentally and will report on the results.