## 1. Introduction

Microring resonators (MRR) have attracted much attention as multifunctional components for signal processing in optical communication systems [1-4]. Recently, due to their fabrication scalability, functionalization and easiness in sensor interrogation, MRR with chip-integrated linear access waveguides have emerged as promising candidates for scalable and multiplexable sensing platforms, providing label-free, highly sensitive and real-time detection capabilities [5-8]. The near-infrared spectral range and, in particular, the 1.5-λm wavelength band is already employed in several bio-/chemical sensing tasks using MRR [9-11].

If gain is incorporated inside the ring, losses (intrinsic absorption, scattering, bend, etc.) can be compensated, filtering and amplifying/oscillating functionalities are combined [12,13] and the sensing potentialities of the device become enhanced [14]. Due to their excellent spectroscopic and solubility characteristics, phosphate glass is a suitable host for rare earth (RE) high doping and Yb^{3+}/Er^{3+}-codoped phosphate glass integrated waveguide amplifiers and lasers provide a compact, efficient and stable performance [15]. However, when the host material of an MRR is Yb^{3+}/Er^{3+}-codoped, the modelling of the performance of the active structure becomes much more complex, since the coupled evolution of the optical powers and the rare earth (RE) ions population densities has to be properly described. Moreover, Er^{3+}-ion efficiency-limiting energy-transfer inter-atomic interactions (homogeneous upconversio n and migration), which are enhanced by high RE-doping levels required by the device dimensions, have to be considered for an optimized design [16].

In the literature, a few models describing RE-doped microfiber ring lasers can be found [17], but there the dopant concentrations level was much lower than those needed in MRRs. Additionally, a simplified model for RE-doped MRR has been proposed where the energy-transfer mechanisms were directly ignored [18]. In a previous paper, we incorporated the effect of high dopant concentrations and presented some results of the optimized active performance of this device [19]. However, in that paper not only the gain coefficient was averaged along the amplifier total length but also the pump resonant behaviour inside the ring and the influence of coupler additional losses were neglected. In subsequent papers we developed a much more detailed model of the performance of a highly Yb^{3+}/Er^{3+}-codoped phosphate glass add-dropfilter filter that overcame previous models deficiencies. An active MRR was described by using a formalism for the intensity rates of the optical powers (pump and signal) at resonance affected by their interaction with the dopant ions through absorption/emission processes. Thus, the performance of an active MRR could be calculated in order to analyse its optimized design and to determine the conditions to achieve amplification and oscillation [20,21].

Drop-port output power maximizing symmetrically coupled structures are mostly used in MRR-based passive components. Alternatively, asymmetric waveguide/MRR coupling may offer definite optimum functional behaviour [22]. For instance, with critically coupled MMRs, the highest throughput attenuation can be attained [23] or when MRR are used as dispersion compensators in the time domain [24].

In this chapter we present a review of our previous works in modelling of Yb^{3+}/Er^{3+}-codoped phosphate microring resonator amplifiers. First, in Section 2, a detailed model of the performance of a highly Yb^{3+}/Er^{3+}-codoped phosphate glass add-drop filter is presented. This model describes the coupled evolution of the rare earth ions population densities and the optical powers that propagate inside the MRR assuming a resonant behaviour inside the ring for both pump and signal powers. In order to exploit the active potentialities of the structure, high dopant concentrations are needed. Therefore, energy-transfer inter-atomic processes are included in the numerical design. The microscopic statistical formalism based on the statistical average of the excitation probability of the Er^{3+} ion in a microscopic level has been used to describe migration-assisted upconversion. Moreover, due to its high solubility for rare earth ions, phosphate glass is considered an optimum host.

In Section 3, the model is used to calculate the performance of an active microring resonator and the more significant parameters are analysed in order to achieve an optimized design. Finally, in Section 4, the model is used to determine the practical requirements for amplification and oscillation in a highly Yb^{3+}/Er^{3+}-codoped phosphate glass MRR side-coupled to two straight waveguides for pump and signal input/output. In particular, the influence of dopant concentration, additional coupling losses, and the structure symmetry are fully discussed.

## 2. Yb^{3+}/Er^{3+}-codoped microring resonator model

### 2.1. Active integrated microring transfer functions

An MRR evanescently coupled to two straight parallel bus waveguides (commonly termed an add-drop filter) is the structure under analysis (see Fig. 1). In our formalism, the add port is ignored since only amplifiers and laser amplifiers are considered. Clockwise direction, single-mode single-polarization propagation is considered. Moreover, the bus waveguides and the MRR are assumed to have the same complex amplitude propagation constant *β* is the phase propagation constant, *α* is the loss coefficient (due to scattering and bend) and *g* is the gain coefficient. This coefficient describes the evolution of the pump/signal mode amplitudes caused by their interaction with the RE ions.

In Fig. 1 r is the microring radius and the central coupling gaps between each waveguide and the ring are *ith* coupler. Therefore, the actual intensity coupling and transmission coefficients are

*Amplitudes at the couplers output ports.* If the input/output complex amplitudes at the couplers ports are denoted as *i=1,2* for the input/through ports at coupler 1 and *i=3,4* for the add/drop ports at coupler 2, respectively) the following scattering matrix relations can be used to describe the exchange of optical power between the waveguides and the MRR:

and the relations between complex amplitudes at the directional couplers ports are:

Moreover, the transmission along the two ring halves is such that

where

*Intensity rates*. From Eqs (8)—(11) the input/output transfer functions of the structure in Fig. 1, that is the rates of the intensities from the input port to the coupler output ports, can be readily obtained as follows:

where

Finally, if the intensity rates are considered at the output ends of the straight waveguides, the amplitude evolution from/to the coupler output ports along the add-dropfilter waveguides has to be also taken into account.

### 2.2. Pump and signal powers evolution inside the active MRR

We assume that the resonance condition (

Then, to determine the intensity rates in Eqs. (12)—(15) not only the passive characteristics of the microring resonator (losses, coupling and transmission coefficients) are required but also pump and signal gain coefficients, which depend on the active MRR working conditions. The evolution of pump and signal powers inside the resonator greatly differ. Whereas signal gain coefficient is habitually be positive even for low pump powers, pump gain coefficient is always negative since pump experiences attenuation along the ring due to absorption by the RE ions.

*Pump intensity enhancement inside the ring*. The pump power that circulates inside the ring is best described using the intensity enhancement factor, *E*, the rate between the confined and the input intensities, which can be evaluated as the average intensity rate^{15}:

(17) |

*Signal transfer functions and threshold gain coefficient.* For a resonant signal the transfer functions (Eqs. (12) and (15)) become

The intensity rate to the through port,

Fulfillment of the CC condition produces the complete destructive interference between the internal field coupled into the output waveguide and the transmitted field in

Finally, if

### 2.3. The Yb^{3+}/Er^{3+}-codoped system in phosphate glass

Due to the large Yb^{3+} absorption cross section in the 980-nm band compared to that of Er^{3+}and the good overlapping between the Er^{3+}-ion absorption spectrum (^{4}I_{15/2}⇒^{4}I_{11/2}) and the Yb^{3+}-ion emission spectrum (^{2}F_{5/2}⇒^{2}F_{7/2}), ytterbium is a good sensitizer to efficiently improve the gain performance of Er^{3+}-doped waveguide amplifiers. Moreover, because of their high solubility for RE ions and their excellent optical, physical and chemical properties, phosphate glasses stand out among all laser materials for RE-doped waveguide amplifiers and lasers. In particular, high dopant concentrations can be achieved without serious ion clustering [25].

The schematic energy level diagram of aYb^{3+}/Er^{3+}-codoped phosphate glass system is shown in Fig. 2. We assume the model for a 980-nm pumpedYb^{3+}/Er^{3+}-codoped phosphate glass waveguide amplifier presented in Ref. [16]. In this model the temporal evolution of the population densities of the levels, ^{3+}/Er^{3+}-codoped system, which can be written as follows:

where the population densities of the ytterbium ion levels ^{2}F_{7/2} and ^{2}F_{5/2}, and of the erbium ion levels ^{4}I_{15/2}, ^{4}I_{13/2} and ^{4}I_{11/2}, are *i*, whereas the values of the densities of stimulated radiative transition rates,

where ^{±}) waves, with optical frequency *ν*. *ith* and *jth* levels. Concerning the energy-transfer inter-atomic mechanisms, a term proportional to the Er^{3+}-ion first excited level population squared is used to describe the upconversion effect. The homogeneous upconversion coefficient (HUC) assesses the number of upconversion events per unit time and is a function of the first excited level population, ^{3+}⇒ Er^{3+} energy transfer and back transfer coefficients are

We use available parameters from measurements on Yb^{3+}/Er^{3+}-codoped phosphate glass in order to numerically evaluate the Yb^{3+}/Er^{3+}-codoped system rate equations. In particular, the fluorescence lifetime of the Yb^{3+}-ion level ^{2}F_{5/2} is assumed to be 1.1 ms [27], that of the Er^{3+}-ion levels ^{4}I_{13/2}and ^{4}I_{11/2}are 7.9 ms[28] and 3.6 x 10^{5} s^{-1} [29], respectively. Both absorption and emission cross-section distributions for the 1535-nm band are taken from Ref. [30], and the 976-nm pump laser cross sections are taken from Ref. [28] for both ions. According to Ref. [16], for weak CW pump, the HUC coefficient is nearly a constant, whereas for high pump range it is a non-quadratic function of ^{3+} ion concentration increases, the upconversion coefficient also increases due to the migration contribution. This formalism was recently adapted to include Yb^{3+}-sensitization and transversally resolved rate equations, which become essential due to the nonlinear character of the energy transfer mechanisms.

Concerning the Yb^{3+} ⇒ Er^{3+} energy-transfer rate, we assume the fitted values to an experimental dependence of the energy transfer coefficient in [32]. Finally, since the population in the Er^{3+}-ion level ^{4}I_{11/2} remains low even at high pump powers, in practice, the value of the Er^{3+} ⇒ Yb^{3+}back transfer coefficient can be assumed as a constant,

### 2.4. Propagation of the optical powers

The evolution along the active waveguide of the pump, signal and ASE powers can be expressed as follows:

In Eqs. (27) and (28), *γ* is *p* for pumping and *s* *s* for signal. The pump and the signal are assumed to be monochromatic and the wavelength-dependent scattering losses are denoted as ^{-4} law. Finally, in Eqs. (27)—(28) the coupling parameters, *ith* level population density distributions over A, which is the active area,

In Eq. (29), A is defined as the area for which the integral of the addition of population densities of the excited levels converges within a required precision. Finally, a Runge—Kutta-based iterative procedure can be used to numerically integrate the equations that describe the propagation of the optical powers along the waveguide, Eqs. (27) and (28).

## 3. Numerical analysis of an active add-drop filter

### 3.1. Passive structure

An air-cladded ridge guiding structure, which presents attractive features for sensing applications [34], has been adopted for the calculations. In Table 2 we summarize the passive parameters of the structure.

The amplitude coupling ratios for pump and signal at each coupler are functions of

When additional coupling losses are included in the model, the practical range of central coupling gap and accordingly of the amplitude coupling ratio will be further limited. For our analysis the range of additional coupling losses is estimated from Ref. [36], where the value 0.014 is obtained for d=117±5 nm. When the coupling gap between the MRR and the access waveguide is below this value they report a significant increase of these losses.

### 3.2. Pump enhancement inside the microring

Besides the lossless amplitude coupling coefficient,

Then, using the equations for the coupled evolution of the population densities and optical powers we have calculated the pump amplitude gain coefficient in a waveguide with L = 97.20 µm (2π x 15.47 µm) as a function of the average circulating pump power. This dependence is plotted in Fig. 5 for five concentration pairs ^{26} ions/m^{3}. RE ions concentration values were chosen with *n*_{Yb}*= 2n*_{Er}, since this rate is often used experimentally. The amplitude pump gain coefficient varies greatly with the average circulating pump power inside the ring. Low pump powers are strongly attenuated as the dopant concentration increases whereas high pump powers are relatively less affected by rare earth absorption.

As shown in Fig. 4 and 5,

### 3.3. Signal gain coefficient

First, as with the pump intensity enhancement, we analyse the dependence of the signal intensity rate between the drop and the input ports,

For each value of

As it can be appreciated in Fig. 7,

By comparing Figs. 6 and 7, the minimum RE ions concentrations necessary to achieve a significant amplification can be estimated as a function of

## 4. Gain/oscillation requirements for a symmetric structure

### 4.1. Net gain requirements for a symmetric structure

Firstly, the requirements to achieve net gain and oscillation are going to be analysed in a symmetric structure and afterwards we extend this analysis to asymmetric structures. Therefore, in Sections 4.1 and 4.2, equal lossless amplitude coupling ratios for both pump and signal powers between the microring and the straight waveguides (

Net gain dependence on

As the additional losses increase, the value of

### 4.2. Threshold gain and oscillation requirements for a symmetric structure

Then, we are going to analyse the oscillation requirements. In Fig. 9, the evolution of the threshold gain as a function of the lossless amplitude coupling coefficient, ^{-1}, 83.1 m^{-1}, 134.9 m^{-1} and 187.0 m^{-1}, respectively. Hence, in order to achieve the necessary

A further optimization of the structure could be accomplished if non-symmetric schemes are considered. In the next section we analyse the gain/oscillation requirements when different values for the lossless amplitude coupling ratios and additional coupling losses between each straight waveguide and the microring are allowed.

## 5. Gain/oscillation requirements for an asymmetric structure

In order to parameterize the asymmetry of the structure, we use the relative variation of the lossless amplitude coupling coefficient,

### 5.1. Asymmetry influence on pump enhancement

Pump enhancement presents a maximum as a function of

In Fig. 10, the evolution of the maxima position and value are represented as a function of

### 5.2. Asymmetry influence on the drop/input port intensity rate, I_{41}

As we did with symmetric structures, net gain for asymmetric MMR is calculated. In Fig. 11, the evolution with

Next, we study the performance in CC conditions. Differently from the passive MRR, in an active structure the value of

In Fig. 12, the values of

### 5.3. Asymmetry influence on threshold gain

Finally, changes in

In Fig. 14 we can see how the necessary threshold gain value decreases for

## 6. Conclusions

In order to optimize RE-doped amplifying/oscillating MRRs, the coupled evolution of resonant pump and signal powers inside the integrated structure must be modelled and the interrelated passive and active characteristics must be taken into consideration. The RE ions concentration sets the attainable signal gain coefficient. This coefficient, together with the pump intensity enhancement dependences, determines the suitable combination of passive parameters (greatly influenced by the expected additional coupling losses) and RE ions doping level to achieve significant amplification or oscillation operation.

A further optimization could be achieved if non-symmetric structures are considered, allowing different values for the lossless amplitude coupling ratios and the additional coupling losses between the microring and the straight waveguides. The use of asymmetric structures can to some extent relieve the demand of a much higher signal gain coefficient and threshold gain (and accordingly dopant concentrations) as the additional losses increase. Structures with lower output coupler coupling coefficient than the input coupler one are preferable. Finally, since signal gain saturation is achieved for relatively low circulating pump powers (due to the short length of the MRR), in practice, asymmetry has little influence on pump enhancement.