End-pumped Nd:YAG laser with a longitudinal hyperbolic dopant concentration profile

Ralf Wilhelm; Denis Freiburg; Maik Frede; Dietmar Kracht

doi:10.1364/OE.16.020106

1. Introduction

Due to a high overlap of the spatial inversion distribution and the resonator’s fundamental eigenmode, end-pumped solid-state rod lasers exhibit a high optical-to-optical efficiency [1]. In conventional end-pumped lasers involving crystals with a homogeneous dopant concentration distribution, the maximum pump power for a given rod of length L is limited by the peak values of the temperature and thermally induced mechanical stress distributions resulting from the exponential decay of the pump light intensity. For a pump absorption efficiency of 95 % and a monochromatic pump source, it turns out that a crystal with a uniformly doped segment would have to be 3 times longer than a transversally pumped crystal with the same power capability [2], if the transversal pump light distribution remains unchanged.

Among the already existing approaches for reducing the crystal length required is the realization of a pump light double pass by providing the crystal with a highly reflective coating for the pump wavelength on one end and pumping it from the other end [3]. In this case, the required length is reduced by nearly a factor of 2 (about 1.9 for 95 % pump absorption efficiency) compared to the single pass case. The same applies when the crystal is pumped simultaneously from both ends.

While these approaches result in simple crystal designs their main drawback is that a significant fraction of the incident pump light is reflected to the pump optics and might endanger the pump source. An alternative eliminating this issue is to employ crystals with longitudinal dopant concentration profiles. Multi-segmented rods approximate the desired longitudinal hyperbolic dopant concentration profile by a piecewise constant function. The techniques to manufacture these rods are well established and crystals are available on an industrial scale, but for a maximum practical segment number of 3 doped segments plus 2 undoped end caps, the maximum applicable pump power is limited to about two thirds of the power that could be applied to a transversally pumped rod (at 95 % absorption efficiency) [2]. Employing a crystal with three doped segments, 407 W of multi-mode laser output power have been demonstrated with an optical-to-optical efficiency of 54 % [4]. Further power scaling of these rod would require a relatively high number of doped segments or longer rod crystals.

A rod with a continuously varying dopant concentration profile is studied in this paper. After summarizing the theoretical foundations for the hyperbolical dopant concentration profile, we report on the characterization of a prototype crystal by spatially resolved absorption and fluorescence measurements before utilizing the crystal in a short multi-mode laser resonator at pump powers in excess of 430W.

2. Theoretical foundations

As has already been mentioned in the introduction, the main bottleneck in scaling rod lasers to higher pump powers is related to the thermally induced tensile surface stress which can be approximated in the case of a transversally homogeneous pump light distribution by [5].

σ = \frac{\sqrt{2} α E η_{h}}{8 π K (1 - ν)} \frac{dP}{dz}

with YOUNG’s elasticity module E, the thermal expansion coefficient α, the POISSON number ν and the heat conductivity K, all material parameters assumed to be temperature—independent. The heat efficiency η _h is the fraction of the pump power being converted into heat due the quantum defect (i. e. energy difference between a pump and laser photon) and non-radiative processes competing with the laser transition, such as AUGER upconversion and cross relaxation [6]. Due to the dipole-dipole character of the underlying inter-ionic interaction mechanisms, these processes are dopant concentration dependent and become more pronounced at Nd concentrations in excess of about 0.7 at. % when they start to result in a reduction of the fluorescence lifetime [7, 8]. dP/dz is the power that is absorbed in a crystal segment of unity length. The fracture stress σ_max of the laser material limits the maximum pump power absorbed per unit length to a maximum value dP/dz|max, which in the case of a transversally inhomogeneous pump light distribution depends on the ratio of the pump spot and rod radii [2].

For end-pumped cylindrical rods with a large length-to-diameter ratio and high-power pump sources exhibiting a low brightness, the propagation of the pump light in the crystal is governed by total internal reflection (TIR) of the light at rod’s barrel surface. As has been described earlier [2], the pump light tends to concentrate around the crystal’s axis and the transversal pump light distribution changes in amplitude but retains its shape upon propagation through the crystal. Therefore, it is assumed that the volumetric heat generation density distribution can be described by the product of two functions, one changing in z and the other one describing the transversal heat generation distribution, which is assumed to be radially symmetric in what follows, i. e.

q (r, z) = \frac{P_{pump} η_{h} (z)}{2 π \int_{0}^{R} dr r f (r)} α_{eff} (z) \exp (- \int_{0}^{z} dz' α_{eff} (z')) f (r),

with α _eff(z) being the effective absorption coefficient of the laser material, given by

α_{eff} (z) = \frac{\int_{0}^{\infty} d λ P_{λ} (λ, z) α (λ, z)}{\int_{0}^{\infty} d λ P_{λ} (λ, z)},

with α being the absorption coefficient and P _λ is the spectral power density of the pump light. The function f (r) may be approximated by a truncated GAUSSIAN distribution. For a crystal with a varying dopant concentration, the heat efficiency as not spatially constant anymore but varies with the dopant concentration. However, assuming a unity pump-metastable quantum efficiency, the variation will be small if the stimulated emission intensity in the laser resonator is a few ten times higher than the saturation intensity of the laser transition [9]. In the following derivation, a constant η _h is assumed. The effect of a non-unity pump-metastable quantum efficiency, which results in a considerable variation of η _h even in the limit of an infinite stimulated emission intensity, is studied in section 4.

If axial heat transport can be neglected, the pump power absorbed per unit length has to be constant:

\frac{d P (z)}{dz} = - α_{eff} (z) \cdot P (z) = const . = - P (0) ξ \Rightarrow P (z) = P (0) (1 - ξ \cdot z) .

In order to fulfill the second equation of (4), the effective absorption coefficient α _eff(z) has to increase hyperbolically in z

α_{eff} (z) = \frac{ξ}{1 - ξ z},

and the constant ξ is identified as the starting absorption coefficient, for which the optimum value is given by

ξ = \frac{{\frac{dP}{dz} ∣}_{max}}{P_{pump}} .

Since nearly all high-power pump sources exhibit a spectral width in the order of a few nanometers and the absorption coefficient of Nd:YAG is strongly varying with the pump wavelength, the spectral composition of the pump light changes as it propagates through the crystal. Therefore, the optimum dopant concentration profile deviates from a hyperbolic profile. This issue is addressed in section 3.1.

3. Crystal characterization

In order to examine the feasibility of the theoretical concepts described in the previous section, a Nd:YAG rod crystal, 3 mm in diameter, with a doped region 27 mm in length and two 7.5 mm long undoped end caps has been manufactured (FEE GmbH). The barrel surface has been polished in order to enable pump light guidance by total internal reflection without excessive scattering losses.

The doped section has been grown employing the BRIDGMAN-STOCKBARGER method [10, 11]. The experimental characterization of the dopant concentration profile will be described in the next two subsections.

3.1. Absorption measurements

Fig. 1. Determination of the dopant concentration profile of the crystal: (a) experimental scheme of the absorption measurement; (b) geometry for determining the average optical path length.

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In order to determine the dopant concentration profile of the rod crystal, a two-step absorption measurement has been performed (Fig. 1(a)). Both measurements involved a tunable, spectrally narrow laser source with high beam quality (Spectra Physics 3900S tunable cw Ti:Sa laser). Since the emission spectrum of this laser is much narrower (spectral width about 15 cm^-1) than the width of the features in the Nd:YAG absorption spectrum around 800 nm, the intensity of a ray is given by BEER LAMBERT’s law and the change of the spectral composition with propagation density can be neglected.

In the first measurement, the relative absorption coefficient has been determined from a measurement of the extinction of a Ti:Sa laser beam crossing the laser crystal perpendicular to its axis. Due to the finite size of the beam, which has been set to about 1 mm such that sufficient averaging over the microscopic scatterers on the crystal’s barrel surface is obtained, it is to be expected that different parts of the beam will experience different optical path lengths, depending on their point of impact. In order to study this effect, a simple geometrical calculation has been performed (Fig. 1(b)). For a ray hitting the rod’s barrel surface at a vertical distance r away from its axis, the propagation length in the crystal is given by

L_{p} = 2 \cdot \sqrt{R^{2} - \frac{r^{2}}{n^{2}}} .

Taking a refractive index of 1.82 for the crystal material, one finds that even an extreme ray with r=R will still experience an optical path length of about 85 % of the length a ray with r=0 would show.

In order to find the average optical path length as function of a GAUSSIAN probe beam of radius w, one has to average the optical path length over the beam’s cross section, i. e., one has to evaluate

{\bar{L}}_{p} = \frac{4}{π w^{2} (1 - \exp (- 2 R^{2} ⁄ w^{2}))} \int_{0}^{R} \int_{0}^{2 π} d ϕ d ρ ρ \exp (\frac{- {2 ρ}^{2}}{w^{2}}) \sqrt{R^{2} - \frac{ρ^{2} \sin^{2} ϕ}{n^{2}}},

where it has been assumed that the beam is stopped by an aperture with a radius equal to the crystal radius. Since the angular integral results in the complete elliptic integral of the second kind, the whole expression has been evaluated numerically. The calculation yielded an average optical path length L̄_p of slightly more than 97 % of the rod’s diameter even for w=R. Since the rays corresponding to short optical path lengths will be at high angles to the optical axis after passing through the rod, these ray will not be detected by a laser power sensor with an active aperture of about 10 mm after a few cm of free propagation. Therefore, it is reasonable to take the rod’s diameter as the optical path length regardless of the probe beam radius.

Fig. 2. (a) Longitudinal dopant concentration profile from absorption measurements and hyperbolic fit. The optimum profiles resulting from solving (10) for different pump wavelengths and absorption efficiencies are shown for comaprison (b) Optimum starting and final dopant concentration values against pump wavelength for 2.5 nm spectral width (FWHM). The measured dopant concentration profile is close to the optimum distribution for a pump wavelength of 802 nm. The absorption efficiency is limited to 80%.

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In the experiment, the transmission of the crystal perpendicular to its optical axis has been measured at a spatial resolution of 0.5 mm. In order to account for the different FRESNEL reflectivity for different rays and the unknown scattering characteristics of barrel surface, a measurement of the transmitted intensity in one of the undoped end caps provided a reference. The measurements have been performed around 808 nm (high absorption) and have been repeated at a wavelength of 800 nm (low absorption) to properly account for lightly and heavily doped regions equally well. Repeating the measurement after rotating the crystal around its axis assured that the unknown properties optical properties of the barrel surface did not influence the results.

Although an absolute dopant concentration profile can be obtained from these measurement, additional confidence has been obtained from determining the extinction of the crystal in longitudinal direction as well. Since the crystal has a highly transmitting coating on both end surfaces, FRESNEL reflection plays no significant role.

The data α(z) taken from the transversal measurement, and the longitudinal transmission T have to fulfill the condition

T = \exp (- \int_{0}^{L} dz α (z)) .

The longitudinal transmission T was 0.22±0.2 at 800 nm, which corresponds very well with the value expected from the independent transversal measurement (T=0.24). At 808 nm, the measurement was not possible due to the very low transmission of the sample at this wavelength.

Once the absolute absorption coefficient has been obtained, the dopant concentration is obtained from the ratio of α(z) and the absorption coefficient at a known Nd concentration (e. g. 1 at. % Nd). A high resolution absorption spectrum has been recorded by FOURIER Transform Spectroscopy. The absolute Nd concentration of the sample used for this measurement was known with an relative accuracy of 1 % [12]. From this spectrum, the expected monochromatic absorption coefficients are 0.77 and 3.98 cm^-1 at 800 and 808 nm, respectively. The dopant concentration profile deduced from these measurements is well described by a hyperbolic function and is given in Fig. 2(a).

For rod of given length L, the optimum starting dopant concentration depends on the pump wavelength, the spectral width of the pump source and the desired absorption efficiency.

The optimum dopant concentration profile C(z) is found by numerically solving

\frac{d P (z)}{dz} = \frac{C (z)}{C_{0}} \frac{\int_{0}^{\infty} d λ α_{0} (λ) P_{λ} (λ, z)}{\int_{0}^{\infty} d λ P_{λ} (λ, z)} = \frac{1 - T}{L}

for a given absorption efficiency 1-T. Here, C ₀ is the dopant concentration at which the absorption spectrum α ₀(λ) has been measured (e. g. 1 %) and P _λ is the spectral power density of the pump light.

Due to wavelength dependence of the absorption coefficient, the optimum dopant concentration distribution deviates slightly from a true hyperbolic profile and is more accurately described by an expression of the form [2]

C_{opt} (z) = \frac{c_{1}}{1 - c_{2} z} + \frac{c_{3} z}{1 - c_{4} z} .

For comparison, the optimum dopant concentrations profiles for different pump wavelengths and absorption efficiencies have been calculated by solving (10) an are shown in Fig. 2. The values for c ₁ to c ₄ in (11) have been obtained by performing a LEVENBERG-MARQUARDT fit to this data and are given in table 1. Due to the relatively low ratio of the starting and the final dopant concentrations, an absorption efficiency of about 80 % is to be expected from this crystal. Because the starting dopant concentration is in the order of 0.4 %, which is about 4 times higher then the optimum value for conventional pumping at 807 nm, the dopant concentration profile would be of no use when pumping at this wavelength. Instead, a pump wavelength between 802 and 803 nm is much more adequate for this crystal (Fig. 2(b)).

Table 1. Coefficients c₁ to c₄ used in (11) for describing the optimum dopant concentration profile. Bold lines are shown in figure 2 a. The fitting parameters used in the temperature calculation to described experimentally obtained data are given in the last line for comparison.

View Table

3.2. Fluorescence measurements

Fig. 3. (a) Experimental for determining the longitudinal fluorescence distribution. (b) Longitudinal fluorescence distribution upon spectrally narrow excitation at 803 nm.

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Apart from being an independent verification of the dopant concentration profile, a measurement of the fluorescence distributions upon excitation of the pump transition by a collimated, spectrally narrow laser beam permits to determine the the influence of interionic coupling on the inversion (“quenching”). The experimental setup for recording the fluorescence profile is shown in Fig. 3(a)). The crystal is mounted such that its barrel surface is visible and the collimated cw Ti:Sa laser beam is used to excite the pump transition. The fluorescence around 1064 nm is spectrally filtered and collected by a photographic lens in order to keep vignetting and optical distortion at a minimum. A CCD camera (WinCamD) is used at the image plane of the lens to record the fluorescence profile. For processing the fluorescence data, the fluorescence intensity from all camera pixels corresponding to the same axial position is summed up. The distribution have been recorded for probe wavelengths in 0.5 nm steps in the range from 798 to 811 nm.

An exemplary fluorescence distribution is shown in Fig. 3(b). Upon calculating the fluorescence distribution that would have to be expected from the dopant concentration profile determined in the previous subsection, it turns out that the calculated fluorescence distribution agrees with the experimental results only for the first 15 mm, corresponding to dopant concentration below about 1 at. %. For higher dopant concentration, the fluorescence intensity determined experimentally is weaker by up to a factor of almost 2 when compared to the theoretical values.

The main mechanism reducing the fluorescence a reduction of the upper laser level lifetime due to interionic cross-relaxation. In this process, a Nd ion in the upper laser state (⁴F_3/2) exchanges energy with a neighboring ion in the electronic ground state. After the energy exchange process, both ions find themselves in an intermediate state (⁴I_15/2) lying at about half the energy of the upper laser level. Assuming that the interaction can solely be described by dipole-dipole interaction, the transition rate would by proportional to the square of the ion density [7]. However, this is only true for dopant concentrations in excess of about 1 at. %. For lower concentrations, a linear dependence of the fluorescence rate on the dopant concentration is observed that can be explained if the actual distribution of distribution of acceptors (positions in the crystal lattice) is taken into account.

Brown [9] has used an expression of the form

τ (ρ) = τ_{rad} \exp (- b ρ^{c})

instead of the DEXTER function for the effective lifetime τ as a function of the dopant concentration ρ. Since it describes the data given in [7] very well, the parameter set τ _rad=260 µs, b=0.0668 (at. %)^-c and c=2.67 obtained by Brown by fitting (12) to the quantum efficiency data measured by Lupei et al. [13] is used. This parameter set results in fluorescence lifetimes shorter than the values given in [8] for dopant concentrations in excess of 1 at. %. which may be a result of a different definition of the effective lifetime in this data.

Assuming a weak excitation of the laser material, depletion of the electronic ground state and AUGER up-conversion [6] can be neglected and the fluorescence intensity is given by

I_{fluor} \propto ρ (z) I_{pump} (z) \frac{τ_{ρ}}{τ_{rad}}

The fluorescence data calculated from (13) agrees very well with the experimental data except for dopant concentrations higher than about 1.5 at.%. This might be due to a dopant concentration dependent pump-metastable pumping efficiency due to relaxation from nonradiative Nd sites [9, 14].

4. Temperature distribution

In order to identify the optimum pump wavelength for the given crystal, the temperature distributions have to be calculated. A full description of the problem would need a non-sequential ray tracing calculation which would require the development of customized ray tracing code in order to implement the calculation of absorbed power densities in a medium with a continuously varying absorption coefficient. An alternative approach could be to approximate the continuous absorption profile by a piecewise constant function.

Since it has been shown elsewhere [2], that the heat generation density distribution in long end-pumped rods may be described by the product of one function depending solely on r and a second function depending solely on z, a simplified model approximating the transversal pump light distribution by an analytical expression in r has been used.

The temperature distribution is given by the solution of the stationary heat generation equation (assuming angular symmetry)

(\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial^{2}}{\partial z^{2}}) T (r, z) = - \frac{1}{K (T)} [q (r, z) + \frac{\partial K}{\partial T} ({(\frac{\partial T}{\partial r})}^{2} + {(\frac{\partial T}{\partial z})}^{2})] .

Fig. 4. (a) On-axis temperature distribution for 440 W of pump power with laser action. Dashed lines with unity pump-metastable quantum efficiency, solid line with dopant concentration dependent pump-metastable quantum efficiency. (b) Maximum applicable pump power before reaching fracture limit of 130 MPa. Solid lines: rod with hyperbolic dopant concentration profile, dashed lines: uniformly doped rod with same absorption efficiency at 803 nm.

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The FOURIER based approach that has been used for solving this equation has been described earlier [15]. The model assumes a transversally GAUSSIAN heat generation density distribution (1.5 mm 1/e ²-radius), a pump power of 440 W and convective heat transfer coefficient of 1 W/K·cm² at the barrel surface. The dopant concentration profile has been approximated by a hyperbola, i. e.

C (z) = \frac{c_{1}}{1 - c_{2} z},

with the values for c ₁ and c ₂ given in the last line of table 1. A spectral width of 2.5 nm has been assumed for the pump source. The temperature dependence of the heat conductivity has been taken from [16] (data for undoped YAG); the weak dependence of K on the Nd concentration that has been reported recently [17] has not been taken into account.

The total heat fraction η _h is not constant but is a function of the pump and laser wavelengths, the inter-ionic quenching rate, the fraction of nonradiative sites and the intensity of the stimulated emission in the laser resonator. Brown [9] has studied the different contributions to heating coefficient in detail and followed his approach to calculate η _h(z). Since the intensity of the stimulated emission in the resonator is about 20 times the saturation intensity of the laser transition, the spontaneous fluorescence is bypassed by the stimulated the stimulated process and values for η _h starting from 0.24 in the zero concentration limit and increasing nonlinearly to 0.26 for a Nd concentration of 2 at. % results if the fraction of nonradiative sites is zero. However, many authors have reported a much higher value of η _h especially for dopant concentrations in the order of 1 at. % and an the absence of of lasing (e. g. [14]). This has been attributed to the presence of nonradiative sites reducing the pump-metastable quantum efficiency. We used a linear function for describing the reduction of this quantity with increasing dopant concentration as has been proposed by Brown. In this case, η _h varies linearly from 0.24 in the zero-concentration limit to nearly 0.49 for a dopant concentration of 2 at. %. The on-axis temperatures for the two cases of a unity and a dopant concentration dependent pump-metastable quantum efficiency are compared in Fig. 4(a).

In order to determine the maximum applicable pump powers as a function of the pump wavelength, the tensile thermally induced mechanical stress on the rod’s barrel surface was calculated for a given pump power. Then, the pump power was changed iteratively for a maximum stress of 130 MPa, corresponding to the lower end of the range of the fracture stress values found in the literature Fig. 4(b). For comparison, the corresponding data for a uniformly-doped rod is given as well. For the same absorption efficiency as the rod with the hyperbolically varying dopant concentration profile, a Nd concentration of 0.67 at. % would be required for the uniformly doped segment. At this dopant concentration, η _h would be about 0.32 in the limit of infinite stimulated emission intensity. This value has been taken for solving the stationary heat conduction equation, with the other parameters unchanged. At 803 nm, the rod with the continuously varying dopant concentration profile beats the rod with a uniformly doped segment by a factor of 1.94, when expressed in terms of absorbed pump power.

5. Experimental results

Fig. 5. Schematic setup of the laser system.

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For evaluating the lasing properties of the crystal, a short plane–plane resonator has been set up around the crystal. The rod has been pumped by a laser diode stack (nLight, type NL-VSA-750-808-F, 8 diode bars). An output power of 436 W at an emission wavelength of about 803 nm, which was found in the previous to be optimum for an equilibrated temperature profile, could be obtained at a coolant temperature of 16 °C.

The radiation from the stack was first coupled by means of a transfer optic into a fused silica rod (60mmlong, 3mmin diameter) for homogenization before a second transfer optic focussed the light into the crystal. Due to the difference of the refractive indexes of the crystal material and the cooling water surrounding the rod, the pump light is guided by total internal reflection at the rod’s barrel surface which has been polished to optical quality.

A plane mirror (AR808/HR1064) and a plane output coupler with an optimized transmission of 10 % formed a short multimode resonator.

At the chosen pump wavelength, only 80 % of the incident pump power has been absorbed in the crystal. At 350 W of absorbed power, 187 W of multi-mode output power have been obtained (optical–to–optical efficiency 53 %), and a slope efficiency of 58 % resulted (Figure 6). These values correspond very well to results obtained earlier with multi-segmented crystals. While an increase of the efficiency is not to be expected from the continuously varying dopant concentration approach, the overall laser performance is not compromised from the relatively high dopant concentration at one end of the rod, which results in a non-unity pump-metastable quantum efficiency.

6. Summary

In summary, power scaling of end-pumped lasers by a continuously varying dopant concentration profile with an hyperbolic shape has been examined theoretically and demonstrated experimentally. With a deposited pump power of 350W, a multi-mode output power of 187W, corresponding to an optical-to-optical efficiency of 53 % has been demonstrated. The slope efficiency of 58 % compares well to end-pumped lasers employing rods with a constant dopant concentration profile. The design combines the high laser efficiency obtained by end-pumping with the power capabilities of transversally pumped lasers.

Fig. 6. Multimode laser output power against absorbed pump power.

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Acknowledgment

This work was partly funded by the German Ministry of Education and Research under contract 13N8299.

References and links

1. S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992). [CrossRef]

2. R. Wilhelm, M. Frede, and D. Kracht, “Power Scaling of End-Pumped Solid-State Rod Lasers by Longitudinal Dopant Concentration Gradients,” IEEE J. Quantum Electron. 44, 232–244 (2008). [CrossRef]

3. M. Frede, R. Wilhelm, M. Brendel, C. Fallnich, F. Seifert, B. Willke, and K. Danzmann, “High power fundamental mode Nd:YAG laser with efficient birefringence compensation,” Opt. Express 12, 3581–3589 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-15-3581. [CrossRef] [PubMed]

4. D. Kracht, R. Wilhelm, M. Frede, K. Duprç, and L. Ackermann, “407 W End-Pumped Multi-Segmented Nd:YAG Laser,” Opt. Express 13, 10140–10144 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10140. [CrossRef] [PubMed]

5. W. Koechner, Solid-State Laser Engineering (Springer, New York, 1996).

6. S. Guy, C. L. Bonner, D. P. Shepherd, D. C. Hanna, A. C. Tropper, and B. Ferrand, “High-Inversion Densities in Nd:YAG: Upconversion and Bleaching,” IEEE J. Quantum Electron. 34, 900–909 (1998). [CrossRef]

7. R. C. Powell, Physics of Solid-State Laser Materials (Springer, New York, 1998). [CrossRef]

8. A. A. Kaminskij, Laser Crystals: Their Physics and Properties (Springer, Berlin, 1981).

9. D. C. Brown, “Heat, Fluorescence, and Stimulated-Emission Power Densities and Fractions in Nd:YAG,” IEEE J. Quantum Electron. 34, 560–572 (1998). [CrossRef]

10. D. T. C. Hurle, Handbook of Crystal Growth, Vol. 2A (North-Holland, Amsterdam, 1994).

11. K. Becker, “Einkristallzüchtung” in Ullmans Enzylopädie der technischen Chemie (Verlag Chemie, Weinheim, 1978).

12. G. Bitz, Investigation of Correlations between the Optical Properties and the Laser Specific Parameters of Laser-Active Solid-State Materials (PhD thesis, Universität Kaiserslautern, 2001). [PubMed]

13. V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, “Energy Transfer Between Nd³⁺ Ions in YAG,” Opt. Commun. 60, 59–63 (1986). [CrossRef]

14. T. Y. Fan, “Heat Generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron . 29, 1457–1459 (1993). [CrossRef]

15. R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), paper MB46.

16. C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1-kW CW Thin Disc Laser,” IEEE J. Sel. Top. Quantum Electron . 6, 650–657 (2000). [CrossRef]

17. Y. Sato, J. Akiyama, and T. Taira, “Novel Model on Thermal Conductivity in Laser Media: Dependence on Rare-Earth Concentration,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (The Optical Society of America, Washington, DC, 2008), paper CtuQ7.

pump wavelength (nm)	abs. eff. (%)	c ₁ (at. %)	c ₂ (at. % cm^-1)	c ₃ (at. % cm^-1)	c ₄ (at. % cm^-1)
802	80	0.373	0.034	0.135	0.298
	90	0.421	0.025	0.174	0.334
	95	0.447	0.011	0.197	0.352
802.5	80	0.306	0.026	0.139	0.297
	90	0.344	0.031	0.175	0.334
	95	0.365	0.018	0.197	0.352
807	80	0.082	-0.011	0.029	0.297
	90	0.092	-0.017	0.036	0.334
	95	0.097	-0.030	0.041	0.352
experiment (15)		0.361	0.312	—	—

End-pumped Nd:YAG laser with a longitudinal hyperbolic dopant concentration profile

Abstract

1. Introduction

2. Theoretical foundations

3. Crystal characterization

3.1. Absorption measurements

3.2. Fluorescence measurements

4. Temperature distribution

5. Experimental results

6. Summary

Acknowledgment

References and links

Cited By

Figures (6)

Tables (1)

Equations (15)

Optics Express