Simulation of the extraction of near diffraction limited Gaussian beams from side pumped core doped ceramic Nd:YAG and conventional laser rods

Martin Ostermeyer; Ingo Brandenburg

doi:10.1364/OPEX.13.010145

Optics Express
Vol. 13,
Issue 25,
pp. 10145-10156
(2005)
•https://doi.org/10.1364/OPEX.13.010145

Simulation of the extraction of near diffraction limited Gaussian beams from side pumped core doped ceramic Nd:YAG and conventional laser rods

Martin Ostermeyer and Ingo Brandenburg

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Martin Ostermeyer and Ingo Brandenburg, "Simulation of the extraction of near diffraction limited Gaussian beams from side pumped core doped ceramic Nd:YAG and conventional laser rods," Opt. Express 13, 10145-10156 (2005)

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Abstract

The application of the recently developed core doped ceramic Nd:YAG rods has the potential to provide better beam qualities compared to conventional rods since the hard aperture of the rod’s boundary can be made wider while the width of the gain region remains the same. Thus, beam truncation and consequential diffraction can be reduced. We apply a finite elements model to calculate the resulting refractive index profiles in conventional and core doped rods. Propagating a Gaussian beam through both rod geometries the impact of aberrations and diffraction is compared for different side pumped scenarios. The potential advantage of the core doped geometry is discussed.

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Fig. 1. Principal sketch for FEA-model.

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Fig. 2. Pump profiles for an absorbed pump power of 378W

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Fig. 3. Refractive index profiles for 378 W of absorbed pump power for Nd:YAG-rod with radius r_rod = 4.5 mm.

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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.

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Fig. 5. Profiles from the caustics in Fig. 4 at different distances z. Rows 1-4 present the corresponding cases 1-4.

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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.

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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.

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Tables (9)

Table 1. Nd:YAG-Parameters used in FEA-Modeling

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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.

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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).

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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.

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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.

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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 r_rod = 1.5 mm, for 378 W of absorbed pump power.

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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.

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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 r_rod = 2 mm, for 378 W of absorbed pump power.

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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 C₄ in the refractive index profiles shown in Fig. 6 and Fig. 7.

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Equations (4)

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n (r) = n_{0} + {C_{2}}^{*} r^{2} + {C_{4}}^{*} r^{4}

n (r) = n_{0} (1 - \frac{r^{2}}{2 n_{0} f L_{rod}})

Φ (r) = \frac{n (r) ∙ L_{rod}}{λ}

E_{i} (r_{2}) = i e^{ik L_{i}} 2 πF \int_{0}^{r_{rod}} E_{in} (r) ∙ e^{iπ F_{i} (L_{i}) (r_{1}^{2} + r_{2}^{2})} e^{i Φ (r_{1})} J_{0} [2 π F_{i} (L_{i}) r_{1} r_{2}] d r_{1}

Parameter	Value
Fractional thermal load	0.241
Thermal conductivity	$K (T) = \frac{\frac{1.9 \times 10^{6} W}{(cm ∙ K)}}{{(\ln ({T ∙ 5.33 K}^{- 1}))}^{7.14}} - \frac{\frac{331 W}{cm}}{T} [8]$
Specific heat at 300 K	5.9 × 10² J/(kg K) [9]
Thin film convection coefficient	1.8 × 10⁴ W/(m² K)

	C₄[m^-4]	C²[m^-2]	n₀	f[cm]	T(r_rod) [K]	T(0)[K]
Case 1	1446	-4.65	1.82255	92.5	294.5	300.4
Case 2	24877	-5.59	1.82255	83.0	294.5	301.0
Case 4	52791	-6.75	1.82256	73.7	294.5	301.9
Case 3	119346	-9.38	1.82259	58.8	294.5	303.4

	C₄[m^-4]	C²[m^-2]	n₀	f[cm]	T(r_rod) [K]	T(0)[K]
Case 1	18790	-15	1.82256	27.5	295.7	301.6
Case 2	266151	-18	1.82257	24.7	295.7	302.2
Case 4	560858	-22	1.82258	21.9	295.8	303.2
Case 3	1263272	-31	1.82261	17.5	295.7	304.6
	$* {(\frac{5}{9})}^{4}$	$* {(\frac{5}{9})}^{2}$		$* {(\frac{9}{5})}^{2}$
Case 1	1790	-4.62		89.1
Case 2	25353	-5.56		80.0
Case 4	53427	-6.79		71.0
Case 3	120339	-9.56		56.7

10°C	C₄[m^-4 ]	C₂[m^-2 ]	n₀	f [cm]	T(r_rod)[K]	T(0)[K]
Case 1	21472	-15	1.82240	28.4	285.7	291.5
Case 2	262019	-18	1.82241	25.5	285.7	292.1
Case 4	548927	-21	1.82242	22.5	285.8	293.1
Case 3	1232954	-30	1.82245	18.0	285.7	294.5
20 °C
Case 1	18790	-15	1.82256	27.5	295.7	301.6
Case 2	266151	-18	1.82257	24.7	295.7	302.2
Case 4	560858	-22	1.82258	21.9	295.8	303.2
Case 3	1263272	-31	1.82261	17.5	295.7	304.6
30 °C
Case 1	16072	-16	1.82272	26.8	305.7	311.7
Case 2	269229	-19	1.82273	24.1	305.7	312.3
Case 4	570540	-23	1.82275	21.3	305.8	313.3
Case 3	1288501	-31	1.82277	17.0	305.7	314.8

	C₄[m^-4]	C₂[m^-2]	n₀	f [cm]
Case 1	148689	-43.0	1.819554	9.75
Case 2	2084599	-51.7	1.819563	8.76
Case 3	9881382	-86.5	1.819602	6.22

	C₄[m^-4]	C₂[m^-2]	n₀	f [cm]
Case 1	148214	-42.6	1.819487	9.82
Case 2	2082286	-51.2	1.819497	8.58
Case 3	9905294	-85.8	1.819535	5.68

	C₄[m^-4]	C₂[m^-2]	n₀	f [cm]
Case 1	46993	-24.0	1.819500	17.5
Case 2	654599	-28.8	1.819509	15.7
Case 3	3102901	-48.2	1.819548	11.2

	C₄[m^-4]	C₂[m^-2]	n₀	f [cm]
Case 1	46429	-23.9	1.819469	17.5
Case 2	656509	-28.7	1.819478	15.3
Case 3	3124836	-48.1	1.819517	10.1

3 mm core	M ² _C₄	M²
W₀ = 1.0 mm	3.2	3.0
W₀ = 1.3 mm	7.5	7.0
W₀ = 1.7 mm	13	12
4 mm core	M ² _C₄	M²
W₀ = 1.0 mm	1.1	1.0
W₀ = 1.3 mm	1.6	1.7
W₀ = 1.7 mm	3.6	4.1
5 mm crystalline	M ² _C₄	M ²
W₀ = 1.0 mm	1.0	1.0
W₀ = 1.3 mm	1.3	1.0
W₀ = 1.7 mm	2.2	1.6

Abstract

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Tables (9)

Equations (4)

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