Deteriorative Phenomena

7.5 Gain in Channel Waveguides Hence, the populations N₁ and N₃ are approximated to be zero.³⁴ Introducing the inver-sion β:=N2/Nt thus leads to:

σ_gain(λ) = β σ_em(λ) (7.28)

The small-signal gain g(λ) obtainable on ⁴F_3/2 → ⁴I_11/2 transitions is calculated for the waveguide channels of the 1.9µm thick Nd(0.5 %):(Gd, Lu)₂O₃ film. It is assumed that due to high pump intensities almost all active ions (N_t= 1.61×10²⁰cm⁻³) are excited to the ⁴F_3/2 level and that the signal intensity is small enough not to affect the resulting inversion ofβ≈1 significantly. According to the values given in Tab. 7.3, the confinement factor Γ is approximated to be 94 %. The resulting gain spectrum is displayed in Fig. 7.13.

The maximum gain G_max of 14.8 dB/cm (at λ= 1079.5 nm) obtainable in the waveguide channels of the Nd:(Gd, Lu)₂O₃ film is significantly higher than the maximum gain of 6.2 dB/cm calculated for the Er³⁺ doped sample.

7 Waveguide Experiments

4I11/2

4S_3/2

4I_15/2 Figure 7.16: Schematic representation of the ⁴I_11/2,⁴I_11/2 → ⁴I_15/2,⁴S_3/2 upconversion process of Er³⁺

Due to these upconversion processes, the inversion and thus the gain is reduced. Further-more, the thermal load of the sample is increased. The resulting change in the relative Stark-level populations and thus the effective cross-sections as well as the resulting spec-tral broadening are usually detrimental to the gain.

The rate of the initial ⁴I_13/2, ⁴I_13/2 → ⁴I_9/2, ⁴I_15/2 upconversion process is k_uc·N_u², with an upconversion coefficient k_uc of approximately 0.25×10⁻²⁵m³/s in Er:Y₂O₃ [Hoe94].

This upconversion coefficient is smaller than the values of at least 1.5×10⁻²⁵m³/s re-ported in [Shi90] for Er:YAG. Since most spectroscopic properties of the lattice matched Er:(Gd, Lu)₂O₃ films and Er:Y₂O₃ bulk crystals have been shown to be comparable, this is also expected for the upconversion coefficient k_uc. Hence, k_uc is assumed to be lower for the lattice matched films than for Er:YAG. In order to obtain the same upconversion rate as in YAG, the Er³⁺ density N_t in the (Gd, Lu)₂O₃ films may thus be slightly higher.

A highly efficient in-band pumped Er:YAG laser was realized with an Er³⁺ concentration of approximately 0.5 % [She06], which corresponds to a doping concentration of approxi-mately 0.25 % in the lattice matched (Gd, Lu)₂O₃ films.³⁵ However, with such low doping concentrations, a maximum gain of merely 2.6 dB/cm is obtainable in the Er:(Gd, Lu)₂O₃ waveguides.³⁶ In order to compensate the high waveguide losses of several dB/cm, a higher doping concentration is most likely required. Therefore, an Er³⁺ concentration of 0.6 % was chosen for first gain and laser experiments in the framework of this thesis. Con-sidering that the upconversion coefficient k_uc is about six times smaller in Er:Y₂O₃ than in Er:YAG, the √

6 times larger density of active ions in the Er:(Gd, Lu)₂O₃ waveguide is justified.

Nd³⁺ Doping

For typical four-level Nd³⁺ lasers with low losses, the population density of the upper laser level is usually clamped at a relatively low value. However, in order to obtain the high gain necessary to compensate the high losses of the PLD films, much higher populations of the metastable ⁴F_3/2 level are required. Due to the waveguide geometry, high pump intensities and thus high inversions can be achieved. The effects of high inversions in planar Nd:YAG waveguides are investigated in [Guy98]. Two energy-transfer processes

35The density of RE ions is approximately 50 % smaller in YAG (1.38×10²²cm⁻³[For99]) than in Y₂O₃.

36This value was calculated as in section 7.5.1, for an Er³⁺concentration of 0.25 % but otherwise identical waveguide parameters.

7.5 Gain in Channel Waveguides are considered, self-quenching of Nd³⁺ by cross-relaxation (see Fig. 7.17), and energy-transfer upconversion (see Fig. 7.18).

4F_3/2

4I_9/2

4I_15/2

Figure 7.17: Schematic representation of the ⁴F_3/2,⁴I_9/2 → ⁴I_15/2,⁴I_15/2 cross-relaxation process of Nd³⁺. This process is usually followed by a decay of both ions to the ground state.

Due to fluorescence quenching of the⁴F_3/2 lifetime, the threshold of Nd:YAG lasers usually increases with high doping concentrations. Therefore, the optimum Nd³⁺ concentration for Nd:YAG lasers is usually in the vicinity of 1 % [Dan73]. Since the ⁴F_3/2, ⁴I_9/2 →

4I_15/2, ⁴I_15/2 cross-relaxation process is depending on the ground-state population, it is expected to have a lower impact at high pump intensities emptying the ⁴I_9/2 manifold.

Thus, higher doping concentrations might be beneficial in a waveguide geometry. Fur-thermore, the threshold power of channel waveguide lasers is usually very low, as high pump intensities can be obtained with relatively low pump powers. Hence, an increase of the laser threshold due to self-quenching is often not as relevant as in bulk lasers. At high population densities of the metastable ⁴F_3/2 level, however, ⁴F_3/2, ⁴F_3/2 → ⁴I_11/2, ²G_9/2 upconversion can become a significant gain-reducing process and limits the Nd³⁺ concen-tration for optimum laser performance.

4F_3/2

4I_11/2

2G_9/2

Figure 7.18: Schematic representation of the ⁴F_3/2,⁴F_3/2 → ⁴I_11/2,²G_9/2 upconversion process of Nd³⁺

The effect of upconversion on the small-signal gain of a Nd(1 %):YAG amplifier is inves-tigated in [Guy98]. In that publication, an inversion β of approximately 0.7 has been calculated for an incident pump intensity five times higher than the saturation intensity

I_sat = h c₀

λ_pτ_uσ_abs(λ_p). (7.29)

Without considering upconversion, an inversion of 0.83 would be obtained for the above

7 Waveguide Experiments

In order to estimate if such large inversions can be obtained in the waveguide channels of the Nd:(Gd, Lu)2O3 film, the saturation intensity atλp= 820 nm and the attainable pump intensity were compared. For the calculation of I_sat, the lifetime τ_u= 230µs measured in section 6.2.2 for the ⁴F_3/2 level was used and the absorption-cross section of the film was roughly estimated to be 2×10⁻²⁰cm², which is approximately half the value given in [For99] for a Nd:Y₂O₃ bulk crystal.³⁷ The calculation results in a saturation intensity of 50 kW/cm². Considering the maximum incident pump power P_max of approximately 400 mW attainable with the Ti:Al2O3 laser used for the Nd³⁺ laser experiments and assuming a homogeneous distribution of the pump power in the approximated channel area of 5µm×2µm, intensities of up to 4×10³kW/cm² can be obtained. Since this intensity is significantly higher than the saturation intensity, a high inversion is possible in the waveguide, even if the pump intensity is considerably reduced due to high coupling and propagation losses.

The attainable small-signal gain is thus most likely comparable to the one determined in section 7.5.1. Therefore, a doping concentration of 0.5 % is expected to be sufficiently high to compensate the waveguide losses. This concentration corresponds to a Nd³⁺ density similar to that of the Nd(1 %):YAG amplifier investigated in [Guy98] and is comparable to the optimum doping concentration given in [Dan73] for Nd:YAG. Hence, a Nd³⁺ con-centration of 0.5 % was chosen for first waveguide laser experiments. However, due to the reduced impact of fluorescence quenching, which is expected for the small waveguide dimensions, higher doping concentrations may be beneficial and should be investigated in future experiments.