## Abstract

A simple analytical expression for threshold pump power in an end-pumped quasi-three-level solid-state laser, which takes into account the influence of energy-transfer-upconversion (ETU), is derived. This expression indicates that threshold pump power can be increased dramatically by ETU, especially in low gain lasers and lasers with pronounced three-level character due to the need for high excitation densities in the upper manifold to reach threshold. The analysis has been applied to an Er:YAG laser operating at 1645 nm in-band pumped by an Er,Yb fiber laser at 1532 nm. Predicted values for threshold pump power as a function of erbium doping concentration are in very good agreement with measured values. The results indicate that very low erbium doping levels (~0.25 at.% or less) are required to avoid degradation in performance due to ETU even under continuous-wave lasing conditions in Er:YAG.

©2009 Optical Society of America

## 1. Introduction

Solid-state lasers utilising simple end-pumped rod architectures have found their way into many commercial laser systems serving a range of applications. The primary attraction of end pumping at low powers is the ability to selectively excite the fundamental (TEM_{00}) mode by matching the deposited pump light distribution to the TEM_{00} intensity profile in the laser medium. This, in turn, allows the use of very compact (low loss) resonators with low threshold pump powers and high optical efficiencies. The availability of relatively high-brightness fiber-coupled diode pump sources allows relatively tight focusing of pump light and hence operation on low gain and/or quasi-three-level laser transitions such as Nd:YAG at 946 nm [1], Er:YAG at 1.6 µm [2], and Tm:YAG at 2 µm [3]. Scaling to higher power levels whilst maintaining high efficiency and good beam quality is quite challenging. The chief difficulty is heat generation in the laser medium and its detrimental impact on laser performance and beam quality [4]. The effects of heat generation are especially pronounced in low gain and/or quasi-three-level lasers due to the need for a high pump deposition density leading, in turn, to a high thermal loading density and strong thermal effects. Other processes, notably energy-transfer-upconversion (ETU), can further exacerbate the situation. ETU is an energy-transfer process between two neighbouring ions in the upper laser level. A donor ion decays to a lower energy level by transferring its energy to an acceptor ion, which is excited to a higher energy level. The acceptor ion then decays from its excited state back to lower energy levels, either radiatively, or, as is most commonly the case, non-radiatively via multi-phonon relaxation. As a result, ETU converts two excited ions into one excited ion in addition to generating extra heat. The net effect is a reduction in the effective upper laser level lifetime and an increase in fractional thermal loading. The detrimental impact of ETU on performance can be particularly severe in pulsed (Q-switched) laser configurations requiring a high upper-laser-level excitation density, or in more complicated (higher loss) cw lasers [5]. Likewise, the requirement for a high excitation density in low gain and/or quasi-three-level lasers (cw or pulsed) can also lead to ETU-induced degradation in performance.

The effect of ETU on performance for a particular laser is governed by the upconversion parameter, the active ion doping concentration and a number of other factors relating to the spectroscopy of the laser transition and the laser design. An accurate knowledge of the upconversion parameter and on how ETU impacts on laser performance is essential for effective laser design, particularly when operating at high power levels.

A number of reports have studied the influence of ETU on laser performance in four-level [6–10] and quasi-three-level lasers [11–13]. The theoretical modeling in these studies [6–15] is based on solving the rate equations for the laser system, and ultimately requires complex numerical calculations to investigate the inter-relationships between the various important parameters, for example ETU, cross-relaxation and re-absorption. The effect of ETU on overall performance can be gauged by determining the effect of ETU on threshold pump power, since the upper laser level excitation density and hence ETU loss is clamped at the threshold level. In this report, we derive a simple analytical expression for threshold pump power in a quasi-three-level laser, which takes into account the effect of ETU. Predicted values for threshold pump power for an in-band pumped Er:YAG laser operating at 1645 nm were compared with experimentally measured values for different Er ion concentrations showing very good agreement, considering only the dominant effect of ETU and, hence, confirming the validity of the expression for, and the need to use, low Er ion concentrations (<0.25 at.%) to avoid significant degradation in laser performance.

## 2. Theoretical model

An analytical expression for threshold pump power can be derived by extending the rate equation approach first introduced by Kubodera and Otsuka [16] to include the effect of ETU. The rate equations for an end-pumped quasi-three-level laser may then be expressed as follows:

where

is the total number of laser photons in the cavity mode, *r _{p}*(

*r, z, t*) is the pump rate per unit volume,

*N*and

_{1}*N*are the ion population densities in the lower and upper laser levels,

_{2}*f*and

_{1}*f*are the Boltzmann occupation factors of the upper and lower Stark sub-levels for the laser transition,

_{2}*ΔN*=

*f*-

_{2}N_{2}*f*is the population inversion density,

_{1}N_{1}*s*(

*r, z, t*) is the laser photon density,

*c*is the speed of light in the laser medium, τ is the fluorescence lifetime of the upper laser level,

_{n}*σ*is the emission cross-section,

*W*is the upconversion parameter and

_{up}*γ*=

*1/τ*, where

_{c}*τ*is the cavity photon lifetime. The above expression assumes that ground-state bleaching is negligible and that the lifetime of the higher energy state to which upconverted ions are raised is much shorter than the lifetime of the upper laser level so that the upconverted ions relax quickly back to the upper laser level (i.e.

_{c}*N*≪

_{3}*N*). We also assume that all the upconverted ions return to the upper laser level. The effects of down-conversion processes, such as cross-relaxation or concentration quenching, could be accounted by introducing a concentration-dependence in the fluorescence lifetime. It is noted that cross-relaxation can be a dominant process for the Er

_{1}, N_{2}^{3+}ion system, particularly at very high doping concentrations and where our model is not longer valid, which in fact, has been exploited to enable the 3 µm laser operation [17].

Under steady-state operating conditions,

Hence, from Eq. (1), we obtain the following expression for the steady-state inversion density *ΔN _{E}(r, z)*,

$$+\sqrt{{(1+2b\tau {W}_{\mathit{up}}{N}_{t}+\genfrac{}{}{0.1ex}{}{{s}_{E}(r,z)}{a{I}_{0}})}^{2}+4{\tau}^{2}{W}_{\mathit{up}}\left({r}_{p}(r,z)-\genfrac{}{}{0.1ex}{}{b}{\tau}{N}_{t}-{W}_{\mathit{up}}{b}^{2}{N}_{t}^{2}\right)}].$$

where *s _{E}(r, z)* is the steady-state photon density,

*a*=

*1/(f*and

_{1}+f_{2})*b*=

*f*=

_{1}/(f_{1}+f_{2}), I_{0}*1/c*=

_{n}στ, c_{n}*c/n, n is the refractive index of the laser rod,*and

*N*=

_{t}*N*+

_{1}*N*is the active ion concentration. Substituting Eq. (5) into Eq. (2) yields the following expression relating steady-state photon number

_{2}*S*to total pump rate

_{E}*R*,

_{p}$$-\underset{\mathit{cavity}}{\int}{s}_{0}(r,z)(1+2b\tau {W}_{\mathit{up}}{N}_{t}+\genfrac{}{}{0.1ex}{}{{S}_{E}{s}_{0}(r,z)}{a{I}_{0}})dV=2a{W}_{\mathit{up}}\gamma {\tau}^{2}{I}_{0}.$$

where *r _{0}(r, z)* and

*s*are normalized distributions defined by

_{0}(r, z)and *R _{p}* is given by

The cavity photon number *S _{E}*, and the pump rate

*R*can be expressed in terms of output power

_{p}*P*and absorbed pump power

_{out}*P*as follows:

_{abs}${S}_{E}=\genfrac{}{}{0.1ex}{}{2{l}_{c}{P}_{\mathit{out}}}{{\mathit{chv}}_{L}T},{R}_{p}=\genfrac{}{}{0.1ex}{}{{P}_{\mathit{abs}}{\eta}_{q}}{h{v}_{p}}=\genfrac{}{}{0.1ex}{}{{P}_{\mathit{in}}{\eta}_{q}{\eta}_{\mathit{abs}}}{h{v}_{p}}$

where *l _{c}* is the effective optical length of the cavity,

*ν*is the laser frequency,

_{L}*ν*is the pump frequency,

_{p}*T*is the transmission of the output coupler,

*P*is the incident pump power on the laser rod,

_{in}*η*is the pumping quantum efficiency,

_{q}*l*is the length of the laser rod,

_{R}*α*is the absorption coefficient for pump light and

_{p}*η*=

_{abs}*1*-exp(-

*α*) is the fraction of incident pump power absorbed in the laser rod.

_{p}l_{R}At threshold, *P _{out}*=

*S*=

_{E}*0*and

*P*=

_{in}*P*, hence Eq. (6) becomes

_{th}$$-\underset{\mathit{cavity}}{\int}{s}_{0}(r,z)(1+2b\tau {W}_{\mathit{up}}{N}_{t})dV=2a{\tau}^{2}{W}_{\mathit{up}}\gamma {I}_{0}.$$

If we assume that the pump beam in the laser medium has a top-hat transverse intensity distribution with waist radius *w _{p}*, and that the laser mode has a Gaussian intensity distribution with waist radius

*w*, then the normalized distributions for pump rate and photon density become:

_{L}Substituting Eqs. (10) and (11) in Eq. (9), and integrating over the laser cavity yields,

where

and *η _{LP}*=1-exp(-

*2 w*) is the spatial overlap factor of the laser mode with the pumped region. The left term in Eq. (12) can be integrated by making the substitution,

^{2}_{p}/w^{2}_{l}*u*=

^{2}*B*exp(-

*α*), giving

_{p}zTo arriving at Eq. (14), we assume that most of the pump light is absorbed, that is, exp(-*α _{p}l_{R}*) ≈0. Recalling that the cavity attenuation coefficient

*γ*=

*1/τ*=

_{c}*cL*, where

_{t}/2l_{c}*L*is the resonator loss parameter,

_{T}*L*=-

_{T}*ln*(

*1-δ*) -

*ln*(

*1-T*), then Eq. (14) can be expressed as

Equation (15) can be readily solved numerically to find the value for *B* and hence, the threshold pump power from Eq. (13) in any given situation. However, for most situations of practical interest *B*<*10*, hence the following approximation can be used:

Using this approximation in Eq. (15), we obtain the following simplified analytical expression for threshold pump power:

If ETU is negligible (i.e. *W _{up}*=0), Eq. (17) reduces to the standard expression for a quasi-three-level laser modified by factor

*η*to account for the different pump beam and laser mode transverse intensity profiles [18]. Equation (17) can be re-written in a simpler form showing how the threshold pump power in the presence of ETU compares to the threshold pump power without ETU as follows:

_{LP}where *F _{q}*=4(

*f*

_{1}+

*f*

_{2})

*σ*/(

*W*). The term

_{up}τα_{p}*F*can be considered as a figure-of-merit for the quasi-three-level laser material. It can be seen that a high figure of merit,

_{q}*F*, is important for maintaining the lowest possible threshold. Highlighting the effectiveness of employing lightly-doped laser materials, as

_{q}*F*has a quadratic dependence on the doping concentration, i.e. through the product of

_{q}*W*and

_{up}*α*. In contrast to the situation for purely four-level lasers, the combined cavity loss term also includes a contribution due to reabsorption from the lower laser level; consequently the value of

_{p}*F*is not the only important factor. Thus, the impact of ETU on threshold for quasi-three-level lasers is generally much stronger than for four-level-lasers. It should also be noted that the impact of ETU on performance depends on the Boltzmann occupation factors of the upper and lower laser Stark sub-levels and hence on temperature. Therefore, for certain quasi-three-level laser media it is evident that the threshold condition may not be achieved, due to cascading effects increasing the crystal temperature through the quantum defect and additional heat loading from the ETU increasing reabsorption loss. This highlights the need for a low active ion concentration and effective thermal management for efficient operation of such lasers.

_{q}## 3. Experiment

In order to investigate the influence of ETU on threshold pump power in a quasi-three-level laser system, we constructed a simple two-mirror Er:YAG laser resonator operating on the ^{4}I_{13/2} to ^{4}I_{15/2} transition at 1645 nm (Fig. 1) [2]. The resonator comprised a plane input coupler mirror (IC) with high reflectivity (>99.8%) for the lasing wavelength at 1645 nm and high transmission (>98.0%) for the pump wavelength at 1532 nm, and a concave output coupling mirror (OC) with a reflectivity of 95% at 1550~1650 nm with radius of curvature, 100mm. Pump light was provided by an Er,Yb co-doped cladding-pumped fiber laser (EYDFL) operating at 1532 nm [19]. The output beam from the EYDFL was focussed to a beam waist radius of ~220 µm inside the Er:YAG crystal. Five Er:YAG rods with doping levels of 0.25 at.%, 0.5 at.%, 1.0 at.%, 2.0 at.% and 4.0 at.% with respective lengths of 58 mm, 29 mm, 15 mm, 7.0 mm and 3.5 mm were tested. The crystal lengths were selected so that all five crystals had approximately the same pump absorption efficiency at low pump powers (i.e. in the absence of ground-state bleaching). The latter was measured to be ~98% indicating that the absorption coefficient in Er:YAG at 1532 nm is ~2.6 cm^{-1}/at.%. The end faces of each rod were antireflection coated to cover the range 1530-1650 nm and the rods were mounted in water-cooled aluminum heat-sinks maintained at a temperature of 17 °C and positioned in close proximity to the plane mirror (IC). The threshold pump power was determined as the power required for the onset of relaxation oscillations, detected with the aid of an InGaAs photodetector and an oscilloscope.

## 4. Results and discussion

Figure 2 shows the measured threshold pump power for the Er:YAG lasers with different Er^{3+} concentrations. For low Er^{3+} concentrations (i.e. 0.25 at.% and 0.5 at.%), the threshold pump power is approximately the same at just over 1 W. However, for higher concentration levels there is a dramatic increase in threshold pump power, up to ~7.8W for the 4.0 at.% doping level. In the case of the 4.0 at.% Er:YAG, lasing was terminated at pump powers slightly higher than threshold suggesting a much higher fractional heat loading than for lower doped crystals. In our experiment, negligible ground state depletion was observed, averaged over the length of the gain medium, by measuring the pump transmission at pump intensities in excess of the measured laser thresholds, for the respective doping levels, in the case where lasing was prevented. Therefore our assumption for negligible ground state depletion is deemed suitable for the cases reported here and is supported by the estimation of the percentage of total ions available that would be necessary to provide enough gain to overcome the total cavity losses (including reabsorption and the output coupling), which is nominally 12% for the cavity configuration described below. Using *f _{1}*=0.022,

*f*=0.21,

_{2}*τ*=6.4 ms [20],

*w*=

_{l}*w*=220 µm, η

_{p}_{q}=0.93,

*σ*=1.9×10

^{-20}cm

^{2}[11],

*δ*=0.01,

*T*=0.05, the calculated threshold pump power without ETU is ~0.9 W, which is in reasonable agreement with the measured threshold power for the Er:YAG rods with low Er

^{3+}concentrations (i.e. 0.25 at.% and 0.5 at.%), but not for higher Er

^{3+}concentration levels. Using a value for the ETU parameter of W

_{up}=3.5×10

^{-18}cm

^{3}/s [21] for a 1.0 at.% Er:YAG crystal, and assuming that for the relatively low concentrations of interest here (i.e. < 5%) the ETU parameter scales linearly with the doping concentration [21] and that the fluorescence lifetime,

*τ*remains constant [20], we can calculate the threshold pump power as a function of the Er

^{3+}concentration by solving Eq. (12) numerically. Figure 2 shows that the predicted and measured threshold pump powers are in close agreement confirming the validity of our model. Values for threshold pump power calculated using the approximate analytical expression Eq. (17) (plotted on Fig. 2) are also in good agreement with experimentally measured values up to a doping level of ~2 at.%, corresponding to a figure-of-merit (

*F*) of 0.08. For doping levels greater than ~2 at.%, the analytical values depart from the numerical calculation non-linearly, this is due to the increased error in the approximation of Eq. (16), and may also be related to enhanced losses associated with the higher thermal load. Hence the analytical expression Eq. (17) can explain the influence of ETU on threshold pump power quite well over the practical range of interest.

_{q}It is clear from Fig. 2 that Er^{3+} doping levels <0.5 at.% are required to avoid significant degradation in performance due to ETU. The disagreement between the calculated and experimental values is attributed to reduction of the effective lifetime due to ETU and cross-relaxation since we used the constant fluorescence lifetime for calculation regardless of a doping concentration.

Figure 3 shows Er:YAG laser output power at 1645 nm versus pump power for 2.0 at.%, 1.0 at.%, 0.5 at.% and 0.25 at.% doping levels. It should be noted that the Er:YAG resonator was designed to allow a straightforward comparison of the rods with different doping levels and hence was optimised for highly efficient operation. It can be seen that, in addition to the increase in threshold pump power with doping level, there is a significant decrease in slope efficiency with increasing doping level. Moreover, there is a pronounced ‘roll-over’ in output power for doping levels of 2 at.% and 4 at.%. We attribute this to additional heat loading generated through the ETU process which results in increased resonator loss (due to aberrated thermal lensing) and an elevated crystal temperature. This increase in temperature will raise the population density in the lower laser level and hence the re-absorption loss. These will increase the upper laser level excitation density required for a round-trip gain of unity and hence will increase ETU and thermal loading. Clearly the increase in thermal loading associated with higher doping concentrations has a very detrimental effect on the performance of the 1.6 µm Er:YAG laser. These results once again confirm the need to use an Er:YAG crystal with a low doping level (<0.25 at.%) with effective thermal management for efficient operation at ~1.6 µm, even when operating in cw mode [2,5,22]. In pulsed (Q-switched) mode of operation the population inversion density is generally much higher than for cw mode hence the impact of ETU on performance will be much more severe unless the measures described above for reducing ETU are adopted.

## 5. Summary

We have developed a simple analytical expression for the threshold pump power in end-pumped quasi-three-level lasers, which includes the influence of ETU. The analysis shows that for active host media that suffer from ETU the threshold pump power can increase significantly with doping concentration. In addition we find that the ratio of the total cavity losses to a figure-of-merit factor, *F _{q}*, provides a simple indication of how much of an effect ETU will have on the threshold of a quasi-three-level laser. As observed in the example reported here, a laser with a 4 at.% Er:YAG crystal, operating on the 1645nm line, had a threshold value nearly an order of magnitude higher than would have been the case with no ETU. Moreover, the increase in thermal loading and hence crystal temperature that results from ETU can have a significant effect on the overall performance, as was demonstrated by the “roll-over” in laser power for the higher concentration Er:YAG crystals investigated here. Not only does ETU reduce the output power extraction efficiency, it is expected that the increased heat loading will also have a detrimental effect on laser the beam quality [4,12]. Very good agreement was observed between the predicted and measured values for threshold pump power as a function of Er

^{3+}concentration for an in-band pumped Er:YAG laser operating at 1645 nm confirming the validity of our model. We conclude that the use of low doping levels and effective thermal management is vital for efficient laser operation and power scaling for this eye-safe transition in Er:YAG and in general for other common quasi-three-level lasers that suffer from ETU.

## Acknowledgments

This work was funded by the Electro-Magnetic Remote Sensing (EMRS) Defence Technology Centre, established by the UK Ministry of Defence. Dr Jacob Mackenzie gratefully acknowledges the support of the Royal Academy of Engineering (RAE) and the Engineering and Physical Sciences Research Council, UK (EPSRC).

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