E. Tanguy[⁺], C. Larat, J.P. Pocholle

[⁺] Faculté des Sciences et des Techniques de Nantes, Groupe de Physique des Solides pour l’Electronique

2, rue de la Houssinière BP 92208 Nantes cedex 03, France. E-mail : tanguy@physique.univ-nantes.fr.

1 Introduction
2 Theoretical description
              2.1 Atomic evolution
      2.2 Photonic evolution
            2.2.1 Beam description
            2.2.2 Equation of evolution.
            2.2.3 Laser output power relation
      2.3 Global equations
3 Continuous wave analysis
      3.1 Numerical analysis
      3.2 Numerical versus experimental results
            3.2.1 Optical pumping by Ti:Al₂O₃ laser
            3.2.2 Optical pumping by a pigtailed laser diode
4 Dynamic analysis
      4.1 Evaluation of the "energy migration time" from ytterbium to erbium
            4.1.1 Ytterbium evolution
            4.1.2 Erbium evolution
      4.2 Q-switch operation
      4.3 Gain-switch operation
      4.4 Comparison with experimental results in Q-switch operation
5 Conclusion

1. Introduction

The erbium glass laser is very attractive, mainly due to its emission wavelength located in the "eye-safe" region [1], in the optical telecommunication window [2], and in a transmission window of the atmosphere. The erbium emission around 1.5 µm has already been demonstrated in the 60’s but these lasers operated only at low temperature due to their 3 level scheme and the use of lamp pumping [3]. But the energy transfer from ytterbium to erbium ions, the realization of codoped matrices [4], and the room temperature operation of high power laser diode emitting at 980 nm make erbium laser very attractive.

The aim of our research is to investigate the dynamical behavior of such a codoped laser, especially in Q-switched and gain-switched mode. For that reason, we have developed a model which is similar to other ones [5][6][7], but none of them have investigated these two switched mode.

The model is based on space dependent rate-equations analysis in which the spatial variations of both the pump beam and the laser beam are taken into account. The model is first validated in CW regime with a good agreement with experimental values. In dynamic resolution, we demonstrate that optimal Q-switch and gain-switch operations are quite difficult to achieve and are very different in this codoped system than in a laser singly doped

In this paragraph, we establish the rate equations governing the different level populations and the photon number in the laser cavity.

2.1 Atomic evolution

The numerical values used in this paragraph are reported in Appendix A.

The pump radiation is absorbed by ytterbium ions (see Figure 1) which have only two levels. Subsequently, the energy is transferred to the erbium level ⁴I_11/2. The back transfer (from erbium to ytterbium) is also possible. The erbium ions which are in the energy level ⁴I_11/2 are de-excited to level ⁴I_13/2 or ⁴I_15/2. The laser transition takes place between the ⁴I_13/2 and the fundamental levels. This laser is a three level one.

Figure 1 : Simplified energy-level diagram of the erbium-ytterbium system.

For simplification, we assume that the emission cross section and the absorption cross section of the laser line are equal. Actually, this fact is true only near the transition peak of the Er ion (1.53 µm). But it’s confirmed by our experimental results : the emission wavelength was always near that peak value.

We also neglect any up-conversion (UC) effects for the following reason :

It is commonly admitted (see for example []) that the main UC process is due to Er-Er cross relaxation : 2 Er ions in the metastable state ⁴I_13/2 interact leading to one in the ground state ⁴I_15/2 and the second in the excited state ⁴I_9/2 (above ⁴I_11/2). The latter then relaxes in a non radiative way back to the metastable state ⁴I_15/2. Therefore, UC is modeled by a relaxation term quadratic in the concentration of the metastable state (). In our case, C @ 10^-24 m³s^-1 (phosphate glasses). Totally excited ions should lead to a UC relaxation rate at least ten times lower than the spontaneous emission rate (1/t ₂₁).

The system is described by the following parameters :

• [m^-3] are respectively the Er³⁺ population densities of the ⁴I_15/2 ground state (lower laser level), of the ⁴I_13/2 upper laser level and of the ⁴I_11/2 level.

• [m^-3] are respectively the Yb³⁺ population densities of the ²F_7/2 level and ²F_5/2 level.

• g _ij [s^-1] is the erbium desexcitation rate from the level i to the level j.

• g _yb [s^-1] is the desexcitation rate from the Yb³⁺ excited state to the ground state.

• s _p [m²] is the Yb³⁺ absorption cross section at the pump wavelength.

• s _e [m²] is the Er³⁺ emission cross section at the laser wavelength.
• s _abs [m²] is the Er³⁺ absorption cross section at the pump wavelength.

• [m^-2s^-1] is the pump photon flux.

• [m^-2s^-1] is the laser photon flux.

• N_er [m^-3] is the Er³⁺ atom density.

• N_yb [m^-3] is the Yb³⁺ atom density.

• k [m³s^-1] is the energy transfer coefficient from the Yb^{3+ 2}F_5/2 level to the Er^{3+ 4}I_11/2 level.
• k’ [m³s^-1] is the energy transfer coefficient from the Er^{3+ 4}I_11/2 level to the Yb^{3+ 2}F_5/2 level.

On the basis of the level scheme (see Figure 1), the following space dependent equations for the Er and Yb population densities can be written [8]:

This system of equations can be simplified with the following assumptions :

• The Yb³⁺ fundamental level is not depleted by the pump (). A pump power density of about 1 kW/cm² is required to obtain N’₃=0.01N_yb.
• The de-excitation from the Er^{3+ 4}I_11/2 level to the Er^{3+ 4}I_13/2 is more probable than the de-excitation from the Er^{3+ 4}I_11/2 level to the Er^{3+ 4}I_15/2 (g₃₁<<g₃₂). Thus we assume g _31» 0.
• The de-excitation from the Er^{3+ 4}I_11/2 level to the Er^{3+ 4}I_13/2 is fast enough to consider the population in the Er^{3+ 4}I_11/2 level in all cases to be almost stationary (secular approximation : ) leading to .
• For the same reason, we also neglect the back transfer process from the Er^{3+ 4}I_11/2 level to the Yb^{3+ 2}F_5/2 level (k’N’₁<< g₃₂).
• The pump absorption by the Er³⁺ ground state is also neglected : . We can justify that point as follows : in stationary mode (from equation (6) leading to the condition : . In our material, the first condition is true because is » 10^-2. The second condition leads to that is also true in our material : N_yb is about 10²¹ cm^-3.
• 

Thus, the atomic evolution is described by the following equations :

2.2 Photonic evolution

2.2.1 Beam description

In this paragraph, we describe the spatial and temporal properties of the laser beam and the pump beam. We make the following assumptions : (i) the laser oscillates in only one cavity mode, (ii) the cavity is linear, (iii) the laser flux is almost constant in the cavity because gain and losses are small (few % in our case).

The photon flux F is equal to the product of the photon density Y in the cavity by the optical wave velocity in the medium :

where n is the optical index of the medium and c₀ is the light velocity.

As a consequence of the third assumption, the intracavity intensity is a weak function of z. Thus, the photon number density can be written as :

where N(t) is the photon number and is the spatial density for one photon in the cavity that is defined by :

Finally, we obtain :

Expression of , both for the laser and pump beam, are presented in Appendix B and C respectively.

In the laser cavity, the laser photon number variation is the photon number created by stimulated emission minus the loss per unit time. The total photon number created by stimulated emission by time unit in the cavity volume V is :

leading to the evolution equation :

where N_L is the photon number in the cavity.

t _cav depends on the cavity losses and the cavity optical length by the relation :

where l_opt is the optical cavity length, R is the output mirror reflection and p is the intrinsic losses per round trip.

It is obvious that if at t=0, N_L=0, the equation (12) leads to N_L(t)=0 at any time t. Actually, we have neglected the spontaneous emission in the laser mode which do act as a photon injection allowing the induced emission ignition. This phenomena can be taken into account simply by considering that the initial photon number is non null (i.e. N_l(0)=1).

The output photon number per time unit is where represent the part of only related to the output mirror transmission : . The output power is thus : where is the energy of one photon and we obtain the relation :

Thus, the laser is described by the three equations :

In this paragraph, we calculate the continuous wave laser output evolution and compare it with experimental results. All the numerical calculations are performed with gaussian shape beams except in § 3.1 below where it is compared with the uniform distribution.

3.1 Numerical analysis

In continuous operation, the atomic evolution becomes :

The resolution of equations (18) and (19) leads to a relation between N₂ and N_L as presented in Appendix D.

These equations are numerically solved :

we fix the output laser power P_l and an arbitrary value of the pump power P_p is chosen. In that conditions if the calculated gain is greater than calculated losses, P_p value is decreased. And the other case P_p value is increased. The calculus then converge to the correct pump value. This operation is repeated for different P_l values to plot the pump laser versus pump power graph.

We can verify these equations for certain limit conditions : if we assume that the Yb^{3+ 2}F_5/2 and Er^{3+ 4}I_13/2 have infinite lifetimes ( and ), then equation (19) leads to : : one absorbed pump photon creates one laser photon by stimulated emission, because there is no longer atomic relaxation. Considering a uniform spatial distribution for the pump and the laser beam (see Appendixes B and C) the threshold power in that case is zero and the efficiency is : . If there is no other losses than the output mirror loss () and if the active material is long enough to absorb all the incident pump power (), the laser efficiency is the quantum efficiency : as expected.

We can also evaluate the maximum efficiency and the threshold power of the laser with a real material and gaussian beams (parameters given in Appendix A). In that case, the input/output relation is linear up to 2 W pumping power (see Figure 2). The 40 W maximum pump power presented in Figure 2 is actually irrelevant as the assumption of §2.1 on the Yb³⁺ depletion failed and as we can expect a fracture of the laser glass at such a high pumping density.

Figure 2 : Laser power versus pomp power for uniform and gaussian beams
no losses – R=99% - w_l=w_p=100µm.

Two different pump sources are investigated : a Ti:Al₂O₃ laser and a pigtailed laser diode. With the first experiment, the intracavity losses (p) and the energy transfer coefficient (k) are adjusted to fit the numerical calculation with the experimental results. These parameter values are then used for the second experiment comparison with no more adjustment.

3.2.1 Optical pumping by Ti:Al₂O₃ laser

A 2 mm long by a 10 mm diameter Yb:Er:glass disc (Kigre QX/Er) is optically pumped by a Ti:Al₂O₃ laser. One face of the disc is HR coated at 1.54 µm and AR coated at 980 nm. The other face is AR coated at 1.54 µm. The resonator design is a plano-plano cavity with an output mirror of 99%. The Ti:Al₂O₃ laser beam is focused with a 200 mm focal length lens (w_p=104 µm) and is considered to be collimated in the active material (Rayleigh range=3.5 cm).

The values of the parameters used are presented in Appendix A. We have adjusted k and estimated p (the only parameters not measured) to obtain a good agreement with experimental results. The Figure 3 presents the best fit between experimental and theoretical results. Because of the low gain, this model is very sensible to the variation of p and k, and we have taken the values of this parameters for which the theoretical results are closest to the experimental results.

Figure 3 : Laser power versus pomp power for experimental and theoretical pumping by a Ti:Al₂O₃ laser.

3.2.2 Optical pumping by a pigtailed laser diode

With the laser diode pumping the FWHM emission width is greater than that of the absorption transition. So, we have measured the effective absorption and divided it by the Yb³⁺ concentration to obtain an effective absorption cross section.

The cavity scheme is the same that for Ti:Al₂O₃ pumping except that the end of the fiber (125 µm in core diameter and 0.48 in Numerical Aperture) is focused in the active material with a magnification of 2.

We have used the same cavity and the same laser material as in the Ti:Al₂O₃ pumping scheme thus, same parameters have been used, excepted for the description of the pump beam and for its absorption. The pump beam is described as a gaussian transverse distribution with a waist w_p and a divergence Q (see Figure 4).

Figure 4 : Pump beam scheme.

The parameters used are given in Appendix A. The Figure 5 presents the laser power versus the pump power for experimental and theoretical results. For more than 500 mW pump power, the experimental curve does not stay linear. In fact, we have measured the absorption coefficient for all points and it decreases when the pump power increases. The results based on the model are close to the experimental results for pump power less than 500 mW.

Figure 5 : Laser power versus pomp power for experimental and theoretical pumping by a pigtailed laser diode.

In this section, we study the dynamical evolution of the laser, both in Q-Switch and Gain-Switch operation.

All the numerical calculations are performed with the uniform distribution (see Appendixes B and C). Equation (12) becomes :

We solve these equations by the Range-Kutta method : with a time increment small enough, we can write: . The function x(t) can be calculated step by step (the cavity round trip time).

One interesting parameter for the dynamical behavior of the laser is the characteristic time (T_er) to achieve the metastable state population saturation. We can think about it also as the "energy migration time" from Yb to Er. An estimation of T_er can be made following two steps : First we consider the evolution of the upper level of the Yb ions (N’₃) without energy transfer to Er ions, leading to an estimation of the N’₃ value achievable in standard case. Secondly, we consider the evolution of the metastable level of the Er ion (N₂) assuming a constant N’₃ population value and no laser action.

4.1.1 Ytterbium evolution

Under continuous pumping and N_er=0, the evolution is exponential :

with and the rise time .

A 1W pump power on an equivalent area for a gaussian waist of 100µm (F _p=7.10²⁵.m^-2.s^-1) leads to and a rise time T_yb=920 µs. We can also point out that the Yb³⁺ ground state population is depleted less than 10%.

4.1.2 Erbium evolution

The evolution with constant N’₃ value and without laser action is evaluated. To know the filling time of the Er^{3+ 4}I_13/2 level (N₂), we consider the ytterbium ion (N’₃) as a constant energy tank (the level N’₃ is not depleted by the energy transfer). In this case equation (16) leads to an exponential solution : with and the rise time .

If >10^-4, we find with a rise time .

If >10^-3 we have , the asymptotical value is N_er and the rise time T_er is . Due to the strong absorption of the ytterbium, such a value for N’₃ is easy to achieve (P_p>15 W/cm²) and thus it is easy to achieve a totally inverted laser transition.

As represented in Figure 6, the filling time of the Er^{3+ 4}I_13/2 level vary drastically with the population of the Yb^{3+ 2}F_5/2 level. In our case, the asymptotical value (Er^{3+ 4}I_13/2 level population) is always N_er. This fact is due to the large ratio between N_yb and N_er ().

Figure 6 : Population rise time t _er versus population of the Yb^{3+ 2}F_5/2 level.

4.2 Q-switch operation

The dynamical behavior of such a laser in Q-switch operation is investigated. The calculation is performed with the uniform distribution for the laser and the pump beams. The material is excited during 8 ms with a 1W pump power and without laser action (the cavity is blocked). The cavity is instantly switch on and the excitation is turned off.

A first laser pulse is obtained which is typical of classical Q-switched laser operation (Figure 7a). But energy stored on ytterbium ions excites the erbium ions, again leading to a succession of pulses (Figure 7b). In fact, this is because the pumping of the Er³⁺ ions was not stopped when the pumping of the Yb³⁺ one is finished. Due to the large ratio between populations (), ytterbium acts as a constant pump for erbium [9]. These oscillations corresponds to the classical relaxation oscillation leading to a pseudo free-running mode (Figure 7c). Finally, as the Yb pumping is, in fact, stopped, the energy tank (N’₃) is emptied with a time constant t () where is the threshold population of the Er³⁺ ground state. This behavior was already used to evaluate the energy transfer coefficient [10]. An exponential decrease is obtained for N’₃ and P_l (Figure 7d) until N’₃ is too low to obtain N₂=N₂(threshold) and the laser stops. After that, N’₃ decreases at the rate () but it is no longer constant as N₂ decrease. The fact that the energy is mainly stored on the ytterbium ions leads to a very poor restitution of this energy in a single laser pulse. So an efficient transfer from the pump energy to the laser pulse needs an optimization of the ratio .

Figure 7 : Temporal evolution of the peak power and population of Yb3+ 2F5/2 level and Er3+ 4I13/2 level (time scales are different in the different graphs).

On an other hand, our case is interesting for high pulse repetition frequency Q-switch lasers. Indeed, in continuous pumping we can achieve a pulse repetition frequency much higher than with no output laser pulse degradation (see Figure 8).

Figure 8 : Output peak power and pulse width versus Q-switch repetition rate.

After the Q-switch pulse, the population N₂ is lower than threshold; N₂ would have to be the same as that before the first pulse to obtain a second Q-switch pulse with the same characteristics. The increase of N₂ is due to the energy transfer from ytterbium to erbium. This transfer decreases the ytterbium population and must be compensated by pumping before the appearance of another Q-switch pulse. Thus the maximum repetition rate is given by the energy transfer rate and the pumping rate and not by the erbium lifetime.

In the same way, single pulse gain-switch operation is also difficult to achieve because, if the pump is switched off after the laser pulse, the remaining energy stored in Yb³⁺ leads again to other pulses. It is possible to decrease the pump pulse width and/or pump power to obtain a small enough ytterbium population so that, after the energy transfer, the ytterbium upper level population would be small enough to prevent another pulse. This would imply an energy transfer time to obtain population inversion very long (proportional to the ytterbium upper level population) and large output pulse width with low repetition rate would be obtained. So, gain-switch operation is out of interest in that case (N_er<<N_yb). It could be interesting if N_er» N_yb, but in that case, the interest of codoping fails.

4.4 Comparison with experimental results in Q-switch operation

The CW laser presented in § 3.2.2 has been Q-switched by insertion of a mechanical chopper in the cavity (see ref [11] for more details). We have modeled the temporal variation of the intracavity losses due to the chopper as a gaussian beam cut-off by a disc with slits rotating at a constant frequency : with f = 133 Hz (rotation frequency), r = 6 cm (radius), e = 500 µm (slit width) and w_l the intracavity waist of the laser beam.

The values of the parameters are the same in § 3.2.2, with a 1 W pump power. The cavity losses p have been adjusted (the losses introduced by the chopper are not negligible in comparison to the cavity losses). The value p=1% corresponds to the best compromise between peak power and pulse width. Nevertheless, the peak power and the pulse width of the first pulse are relatively different (40 % and 30 %, respectively) and the second pulse is experimentally about 3.3 µs after the first and theoretically about 6.2 µs after the first. In comparison, the experimental output energy is very well fitted by theoretical ones (see Figure 9). These differences between experimental and theoretical results are probably due to the approximation of the temporal variation of the losses and also to the approximation of the beam shape (uniform model). Nevertheless, the order of magnitude of the peak power and the pulse width is good and the model provides a quite accurate estimate of the energy delivered by the laser.

Figure 9 : Temporal profiles of experimental and theoretical pulses.

 

We have developed a modelisation of the Yb:Er glass laser system. This model is in good agreement with experimental results in steady state, both in Ti:Al₂O₃ pumping and fiber coupled laser diode pumping. The agreement in Q-switch mode is only qualitative. Nevertheless, tendencies are outlined due to the particular dynamical behavior of a codoped system. Especially we pointed out the possibility of achieving Q-switch operation with pulse repetition frequency greater than the spontaneous de-excitation rate of the erbium upper level without output laser pulse degradation. In other hand, gain-switch operation is not well suited for such codoped systems.

In this appendix, we give the parameters used for the different simulations.

• Materials :

g ₂₁=113.6 s^-1 (measured) N_er=1,4.10²⁵ m^-3 (measured)

materials index n=1,5 k=7,1.10^-21 m³s^-1 (adjusted)

c₀=3.10⁸ ms^-1 s _e=7.10^-25 m² (measured)

g _yb=1000 s^-1 (measured) s _p=1,3.10^-24 m² (measured)

N_yb=1,6 10²⁷ m^-3 (measured).

• Cavity :

Air length l_air=0

w_p=w_l=100 µm

P=0

s _p=6.10^-25 m²

w_p=110 µm

w_l=80 µm

Q =16°

The shape of laser beam is formalized for the numerical resolution. Two spatial distribution of the laser flux are considered. A uniform one for the sake of simplicity and a gaussian one more realistic but more TPU consuming. The axe z is in the direction of the propagation and x and y are perpendicular each other and perpendicular to z.

• Uniform distribution

• Gaussian distribution

The shape of the pump beam flux () is formalized for the numerical resolution. As for the laser beam description, we consider two cases.

• Uniform distribution

• Gaussian distribution

The analytical solution of the equation set (18-19) for the population N₂ is presented.

With : , and

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