## Abstract

The response of an Er/Yb-codoped waveguide ring laser to a sinusoidally modulated pump power is studied. Experimentally, resonance peaks are observed and their dependences on the average pump power and the modulation index are analyzed. For high modulation indexes bistable behaviour is found. Numerically, a good agreement is obtained for the resonance peak frequencies by using a straightforward approximate model and assuming a dependence on the average pump power of the macroscopic Yb⇒Er energy-transfer coefficient. This dependence can be related to these mechanisms’ performance for high doping and pump levels when examined in a microscopic statistical formalism.

© 2007 Optical Society of America

## 1. Introduction

Erbium-doped waveguides are nowadays presenting a great technological interest because of their use as amplifiers in high functionality integrated structures and as cw or pulsed tuneable laser sources for optical communications, optical storage, medical applications, etc [1]. In particular, Er/Yb-codoped waveguides in phosphate glass offer an excellent performance for high gain optical amplifiers [2]. These waveguides short length imposes high rare-earth doping levels and the concentration-dependent inter-atomic energy-transfer interactions (homogeneous upconversion, sensitisation of Er with Yb, migration, etc.) are favoured. However, the rigorous microscopic statistical modelling of these mechanisms’ performance is extremely involved [3] and usually approximate macroscopic formalisms are used.

The design and optimisation of these devices require a great characterization effort in order to accurately determine their characteristic parameters. During the last years our group has developed new active waveguides characterization techniques in steady state regime, both for transversally accessible waveguides [4,5] and for packaged waveguides [6]. Recently, we are developing new techniques based on the measurement of the dynamical behaviour of the system [7]. These techniques can be more accurate because they do not require the measurement of absolute optical powers and insertion losses. However, their theoretical description involves a much higher complexity and numerical analyses based on approximate models have to be used mostly.

In this paper, we study the non linear response of the Er/Yb-codoped waveguide ring laser to a sinusoidally modulated pump power (resonance peaks, bistability, etc.) and the dependence of the resonance-peak frequency on the average pump power and the modulation factor. Then, a straightforward approximate model is introduced for the system and used for the numerical analysis of the measurements. By assuming a dependence on the average pump power of the macroscopic non radiative Yb⇒Er energy transfer coefficient, a good agreement is obtained for the resonance peak frequencies.

## 2. Experimental

#### 2.1 Set up

For the experiments a channel waveguide was used. The waveguide length is L=5.5 cm, the waveguide section is 6×4 µm^{2}, the distance from the waveguide axis to the glass surface is 7 µm and the refractive index change at the peak of the index profile is approximately 0.04. The buried waveguide was fabricated by Teem Photonics^{™} using a two-step ion exchange process in a phosphate glass codoped with Er/Yb ions (Er^{3+} and Yb^{3+} concentrations are *n _{Er}*=2.0×1026 ions/m

^{3}and

*n*=2.2×10

_{Yb}^{26}ions/m

^{3}, respectively). The waveguide is pigtailed in order to allow an efficient input/output coupling of the optical powers.

The Er/Yb-codoped waveguide ring laser scheme is shown in Fig. 1. A 980-semiconductor pump laser diode was used and two 980/1550 wavelength division multiplexers (WDMs) allowed the incoupling of the pump and the outcoupling of the remaining pump. The laser ring configuration consisted of an active medium (the previously described Er/Yb-codoped phosphate glass waveguide), an optical isolator that causes that the only allowed laser emission will be counterpropagating referred to pump propagation, a tunable bandpass filter to select the laser wavelength, λ_{l}, and finally, to close the ring configuration, a variable coupler was arranged.

#### 2.2 Measurements

All the components in the set up were carefully calibrated (WDMs, connectors, optical isolator, tunable bandpass filter, variable coupler, etc.). The total passive ring length is *l*=30,04±0.03 m and the ring transmission coefficient was measured to be T=0.33 for λ_{l}=1534 nm. In order to carry out the measurements, the intensity current which feeds the pump laser was sinusoidally modulated. The pump laser output power was confirmed to be linear with regard to the modulated current and, accordingly, can be written as: *P _{p}*(

*t*)=

*P*[1+

_{av}*m*cos(2

*πf*)], where

_{e}t*P*is the pump power average value,

_{av}*m*is the modulation index and

*f*is the excitation frequency. The ring laser output signal detection was performed by a PIN photodiode connected to a digital oscilloscope, and the maximum voltage difference in a time period (from now on the peak to peak laser power amplitude) was stored.

_{e}First, the peak to peak laser power amplitude was registered as a function of *f _{e}* for several values of

*P*and for a constant modulation index,

_{av}*m*=0.2. In Fig. 2(a), it can be noticed how as

*P*increases the resonance peaks shift towards higher frequency values and also how the discontinuity in the left wing of the curves progressively vanishes.

_{av}Secondly, the peak to peak amplitude was registered as a function of the excitation frequency for varying *m*, while the average pump power was adjusted in order to obtain a fully modulated laser signal. Therefore, the experimental variation range for *m* (0–0.45) is, in practice, limited by the threshold pump power (137 mW) and the maximum available pump power at the waveguide input end (356 mW for a 900 mA-drive current). In Fig. 2(b), the ascending sense sweeps are represented for 4 values of the modulation factor. Besides, for high modulation indexes other resonance peaks show up (see Fig. 2(b)). According to ref. 8, the reason for the appearance of these amplitude peaks is that as the modulation index increases, the ring laser system favours the development of signal harmonics that fall within the bandwidth around the natural frequency of the system.

Finally, for high modulation indexes, we have observed that the peak to peak amplitude depends on whether the frequency sweep is ascending or descending, so that bistable regions can be found. Fig. 3 illustrates this bistable behaviour for m=0.45. Both ascending and descending sense curves present a discontinuity in the left wing of the resonance peak, but shifted 0.8 kHz, approximately.

## 3. Theoretical

#### 3.1. Model

For the theoretical analysis of the system we assume the model described in Ref. 7. This model is based on the rate equations of the population densities of the involved levels, (*n _{1}*,

*n*,

_{2}*n*and

_{3}*n*are the population densities of the ytterbium levels

_{4}^{2}F

_{7/2}and

^{2}F

_{5/2}and of the erbium levels

^{4}I

_{15/2}and

^{4}I

_{13/2}, respectively) and the equations that describe the propagation of the optical powers along the laser ring (only the pump power and the counterpropagating fluorescence power at the laser wavelength are considered). In order to suppress the transversal coordinates dependence, this model uses the overlapping factors formalism [9] and furthermore, z-independent effective population densities are introduced. Therefore, taking to account that

*n*+

_{1}*n*=

_{2}*n*and

_{Yb}*n*+

_{3}*n*=

_{4}*n*, the equations that describe the temporal evolution of the relative population densities,

_{Er}*n*=

_{2r}*n*/

_{2}*n*and

_{Yb}*n*=

_{4r}*n*/

_{4}*n*, and of the laser power,

_{Er}*P*, in our system are:

_{l}where,

$${T}_{3}={R}_{23}{C}_{23}{n}_{\mathrm{Er}}$$

$${S}_{1}=\frac{{\sigma}_{35}\left({\nu}_{p}\right){\eta}_{3,p}}{h{\nu}_{p}}{P}_{p}\left(t\right),{S}_{2}=-\left(\frac{{\sigma}_{35}\left({\nu}_{p}\right){\eta}_{3,p}}{h{\nu}_{p}}{P}_{p}\left(t\right)+{A}_{43}\right),{S}_{3}=\frac{{\sigma}_{34}\left({\nu}_{l}\right){\eta}_{3,l}}{h{\nu}_{l}},$$

$${S}_{4}=-\left(\frac{{\sigma}_{34}\left({\nu}_{l}\right){\eta}_{3,l}}{h{\nu}_{l}}+\frac{{\sigma}_{43}\left({\nu}_{l}\right){\eta}_{4,l}}{h{\nu}_{l}}\right),{S}_{5}=-{R}_{44}{C}_{44}{n}_{\mathrm{Er}},$$

In Eqs. (4–6), *A _{ij}* represents the spontaneous radiative relaxation rate between the

*ith*and

*jth*levels,

*C*is the homogeneous upconversion coefficient,

_{44}*C*is the non radiative Yb⇒Er energy transfer coefficient, σ

_{23}*is the cross section corresponding to the transition between the*

_{ij}*ith*and

*jth*levels, α

_{i}are the scattering losses at the laser wavelength,

*ν*is the frequency of the optical signals and the label

_{γ}*γ*is

*p*for the pump and

*l*for the laser, L is the waveguide length,

*D*=

*nl*is the passive ring optical path,

*n*being the refractive index, and T is the ring transmission coefficient. Moreover,

*η*are the overlapping factors between the

_{iγ}*ith*level population density and the

*P*intensity modal distributions, and

_{γ}*R*are the overlapping factors between

_{ij}*ith*and

*jth*levels population density distributions. Finally,

*h*is Planck constant and

*c*is the speed of light in vacuum. All the values of the parameters have been taken from Ref. 6., and the overlapping factors have been computed for laser steady state regime and for the average pump power, and have been assumed to remain constant when the pump power is modulated. Since no analytical solution is achievable for the system of equations (1–3) we have to proceed numerically. Switch off initial conditions are assumed for the first frequency of the series and the laser response is calculated by solving the above equations by the fourth order Runge-Kutta method. Then,

*f*is varied over the whole range of interest and for each frequency the end values of the preceding

_{e}*f*are taken as initial conditions. The difference in the initial conditions allows that the laser response for a given frequency may differ, depending on whether the modulation frequency is swept in ascending or descending sense.

_{e}#### 3.2. Results

As it was suggested in Ref. 7, our preliminary results also indicate that to numerically reproduce the experimental resonance peak frequencies for high pump powers, a larger value of the coefficient *C*
_{23} is necessary. Besides, in Ref. 3, where a statistical microscopic formalism is used to describe homogeneous upconversion and migration in an erbium doped silica fiber, it is concluded that for high density and high pump levels the concentration-dependent inter-atomic energy-transfer interactions are more efficient than what the usual macroscopic models describe. Therefore, it seems reasonable to keep on using the more simple macroscopic formalism while introducing a pump-power dependent Yb⇒Er energy transfer coefficient that comprises the increasing efficiency of this mechanism for high pump levels. The value of this effective coefficient, *C*
_{23}, is referred to the value of the non radiative Yb⇒Er energy transfer coefficient determined in Ref. 6 in steady state regime, *C*
^{o}
_{23}=1.8×10^{-23} m^{3}/s, through an average pump-power dependent energy-transfer factor, *f*
_{23}=*f*(*P _{av}*), so that

*C*

_{23}=

*f*

_{23}

*C*

^{o}

_{23}. We checked, as it is indicated in Ref. 7 for the laser transient parameters analysis, that the influence of the value of

*C*

_{44}on the resonance peaks frequencies is, in practice, negligible.

In Fig. 4, we plot the experimental values of the resonance peak frequencies and the numerical curves for 4 values (1, 5, 10 and 20) of the energy transfer factor, *f*
_{23}. From Fig. 4 it is clear that, as *P _{av}* increases, in order to numerically fit the experimental results, higher values of

*f*

_{23}are required.

Therefore, we have calculated the values of *f*
_{23} that fit the resonance peak frequencies measured as a function of *P _{av}* for m=0.2 (see Fig. 2(a)), and used them to compute the resonance peak frequency dependence on the modulation factor

*m*(some of the measured resonance curves are shown in Fig. 2(b)). In Fig. 5 the experimental values are plotted together with the numerical results using the best-fit

*f*

_{23}values.

An excellent experimental/theoretical agreement is obtained. Only for the highest *m* values the numerical and the experimental results start to separate.

## 4. Conclusions

Using an approximate model we have achieved a good agreement between the measurements of the non linear response of the Er/Yb-codoped waveguide ring laser to a sinusoidally modulated pump power and the numerical results. The agreement is obtained by introducing a coefficient for the non radiative Yb⇒Er energy transfer mechanism which depends on the average pump power. This dependence follows the results obtained when microscopic statistical formalisms are applied to these mechanisms. However, a further theoretical analysis is still necessary to fully clarify their real performance in the Er/Yb system and the required large enhancement of the energy transfer coefficient for high pump levels indicates that some other non-conventional energy transfer processes should also be explored.

## Acknowledgements

This work was supported by the Comisión Interministerial de Ciencia y Tecnología, project FIS2006-03639.

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