Parameters used for the gain calculations
A precise model to numerically analyse the performance of a highly Yb3+/Er3+-codoped phosphate glass microringresonator (MRR) is presented. This model assumes resonant behaviour inside the ring for both pump and signal powers and considers the coupled evolution of the rare earth (RE) ions population densities and the optical powers that propagate inside the MRR. Energy-transfer inter-atomic processes that become enhanced by required high-dopant concentrations have to be carefully considered in the numerical design. The model is used to calculate the performance of an active add-dropfilter and the more significant parameters are analysed in order to achieve an optimized design. Finally, the model is used to determine the practical requirements for amplification and oscillation in a highly Yb3+/Er3+-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.
- Active integrated microring resonators
- Yb3+/Er3+-codoped glass
- energy-transfer inter-atomic mechanisms
- gain/ oscillation requirements
- asymmetric structures
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 . Due to their excellent spectroscopic and solubility characteristics, phosphate glass is a suitable host for rare earth (RE) high doping and Yb3+/Er3+-codoped phosphate glass integrated waveguide amplifiers and lasers provide a compact, efficient and stable performance . However, when the host material of an MRR is Yb3+/Er3+-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, Er3+-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 .
In the literature, a few models describing RE-doped microfiber ring lasers can be found , 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 . In a previous paper, we incorporated the effect of high dopant concentrations and presented some results of the optimized active performance of this device . 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 Yb3+/Er3+-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 . For instance, with critically coupled MMRs, the highest throughput attenuation can be attained  or when MRR are used as dispersion compensators in the time domain .
In this chapter we present a review of our previous works in modelling of Yb3+/Er3+-codoped phosphate microring resonator amplifiers. First, in Section 2, a detailed model of the performance of a highly Yb3+/Er3+-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 Er3+ 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 Yb3+/Er3+-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. Yb3+/Er3+-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 . In this expression,
In Fig. 1 r is the microring radius and the central coupling gaps between each waveguide and the ring are . Lossless intensity coupling and transmission coefficients at coupler are and , satisfying . Correspondingly, and are the lossless amplitude coupling and transmission coefficients. Realistically, we also consider additional coupling losses at the waveguide/microring couplers. Even small additional coupling losses may have a large influence on the MRR performance . denotes the coefficient for additional intensity loss at the
and the relations between complex amplitudes at the directional couplers ports are:
Moreover, the transmission along the two ring halves is such that
where . Finally, if we assume that the only input signal is in the input port, the amplitudes at the output ports can be straightforwardly derived as:
where is the length of the ring and is the round-trip gain/loss. Mathematically, this structure is analogous to the classical Fabry—Perot interferometer. The output intensities at the through and drop ports correspond to its reflected and transmitted intensities, respectively. If the couplers are lossless, that is , and there is no ring roundtrip loss, , we obtain . Moreover, if in Eq. (12) , and hence , we obtain the through intensity rate of an all-pass ring resonator with only one coupler:
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 (, where m is an arbitrary integer) is fulfilled for both the pump and signal wavelengths and analyse the evolution of the pump and signal powers inside the MRR.
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.
The intensity rate to the through port, , cancels when the critical coupling (CC) condition is verified:
Fulfillment of the CC condition produces the complete destructive interference between the internal field coupled into the output waveguide and the transmitted field in and, as a consequence, the transmitted intensity drops to zero. From Eq. (18), it can be concluded that if (i.e. ),intensity rates and may be greater than unity and the device is a MRR amplifier. On the other hand if gain compensates all the roundtrip losses and the denominators in Eq. (18) approach zero, and tend to infinity and the oscillation condition is reached. The threshold gain coefficient, , can be calculated as:
Finally, if , the MRR behaves as a laser amplifier. Therefore, the fulfillment of the oscillation condition depends on the achievable signal gain coefficient, what forces a previous optimizing design based on the active MRR working conditions.
2.3. The Yb3+/Er3+-codoped system in phosphate glass
Due to the large Yb3+ absorption cross section in the 980-nm band compared to that of Er3+and the good overlapping between the Er3+-ion absorption spectrum (4I15/2⇒4I11/2) and the Yb3+-ion emission spectrum (2F5/2⇒2F7/2), ytterbium is a good sensitizer to efficiently improve the gain performance of Er3+-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 .
The schematic energy level diagram of aYb3+/Er3+-codoped phosphate glass system is shown in Fig. 2. We assume the model for a 980-nm pumpedYb3+/Er3+-codoped phosphate glass waveguide amplifier presented in Ref. . In this model the temporal evolution of the population densities of the levels, , is described by the rate equations for the Yb3+/Er3+-codoped system, which can be written as follows:
where the population densities of the ytterbium ion levels 2F7/2 and 2F5/2, and of the erbium ion levels 4I15/2, 4I13/2 and 4I11/2, are , , , and , respectively. Notice that, for the sake of simplicity, in Eqs. (21)—(25), the spatial dependence of the population densities and the densities of stimulated radiative transition rates is omitted. Furthermore, and denote the homogeneous ytterbium and erbium ions concentrations. In Eqs. (21)—(25), represents the spontaneous relaxation rate from level
where is the normalized mode envelope  of the pump, signal or co- and counter-propagating amplified spontaneous emission (ASE±) waves, with optical frequency
We use available parameters from measurements on Yb3+/Er3+-codoped phosphate glass in order to numerically evaluate the Yb3+/Er3+-codoped system rate equations. In particular, the fluorescence lifetime of the Yb3+-ion level 2F5/2 is assumed to be 1.1 ms , that of the Er3+-ion levels 4I13/2and 4I11/2are 7.9 ms and 3.6 x 105 s-1 , respectively. Both absorption and emission cross-section distributions for the 1535-nm band are taken from Ref. , and the 976-nm pump laser cross sections are taken from Ref.  for both ions. According to Ref. , for weak CW pump, the HUC coefficient is nearly a constant, whereas for high pump range it is a non-quadratic function of and saturates at the kinetic limit in the case of infinite pump power . As the Er3+ ion concentration increases, the upconversion coefficient also increases due to the migration contribution. This formalism was recently adapted to include Yb3+-sensitization and transversally resolved rate equations, which become essential due to the nonlinear character of the energy transfer mechanisms.
Concerning the Yb3+ ⇒ Er3+ energy-transfer rate, we assume the fitted values to an experimental dependence of the energy transfer coefficient in . Finally, since the population in the Er3+-ion level 4I11/2 remains low even at high pump powers, in practice, the value of the Er3+ ⇒ Yb3+back transfer coefficient can be assumed as a constant, .
|Signal wavelength||1534 nm|
|Pump wavelength||976 nm|
|Decay rate of Yb3+2 ||909 s-1|
|Decay rate of Er3+4 ||127 s-1|
|Decay rate of Er3+4 ||3.6 x 105 s-1|
|Absorption cross section Yb3+2F7/2 at ||10.9 x 10-25 m2|
|Emission cross section Yb3+2F5/2 at ||11.6 x 10-25 m2|
|Absorption cross section Er3+4I15/2 at ||1.5 x 10-25 m2|
|Emission cross section Er3+4I11/2 at ||9.6 x 10-26 m2|
|Absorption cross section Er3+4I15/2 at ||5.4 x 10-25 m2|
|Emission cross section Er3+4I13/2 at ||5.3 x 10-25 m2|
|Energy transfer rate Er3+⇒ Yb3+|
(4I11/2 + 2F7/2⇒4I15/2 + 2F5/2 )
|1.5 x 10-22 m3/s|
|Upconversion critical radius||9.95 Å|
|Ratio between critical radii||601/6|
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), are the optical powers, where z is the distance along the waveguide axis and the label
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 , has been adopted for the calculations. In Table 2 we summarize the passive parameters of the structure.
|Waveguide cross section||1.5 µm x 1.5 µm|
|Substrate refractive index||1.51|
|Core refractive index||1.65|
|Pump wavelength||976 nm|
|Signal wavelength||1534 nm|
|Pump mode confinement factor||0.962|
|Signal mode confinement factor||0.757|
|Microring radius||15.47 µm|
|Pump wavelength resonant order||156|
|Signal wavelength resonant order||96|
|Propagation loss amplitude coefficient||0.25 dB/cm|
The amplitude coupling ratios for pump and signal at each coupler are functions of . In Fig. 3, we plot the ratios evaluated according to Ref. . Particularly, for the more confined pump power a limited range of values is available.
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. , 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, , and the additional pump coupling loss, , the pump intensity enhancement factor, , is basically determined by the pump amplitude gain coefficient, , which reflects the attenuation induced on the pump power by the stimulated transitions in the RE ions. In Fig. 4, is plotted as a function of for 5 values of with . As the pump is more attenuated (the absolute value of increases), the maximum diminishes and is obtained for larger .
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 where concentration units are 1 x 1026 ions/m3. RE ions concentration values were chosen with
As shown in Fig. 4 and 5, depends on the circulating pump power but, in its turn, is a function of . In practice, for given concentration values, if the required average circulating pump to achieve a signal gain coefficient value is calculated, the associated can be determined, and subsequently, the pump intensity enhancement and the necessary input pump power.
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, , on the lossless coupling and on the signal gain coefficient. In Fig. 6, is plotted as a function of the signal gain coefficient for four values of the lossless amplitude coupling coefficient and .
For each value of , does not grow significantly until approaches the threshold gain (when tends to infinity). Then, the input signal is strongly amplified and the rate of growth of is higher for lower . Over the gain threshold laser operation is achieved. As we did with , we now calculate as a function of the circulating pump power for five pairs of dopant concentrations.
As it can be appreciated in Fig. 7, saturates for relatively low circulating pump power for any RE concentration pair. This is caused by the short MRR length, which is much shorter than the waveguide amplifier optimal lengths for each pump power and RE ions concentrations.
By comparing Figs. 6 and 7, the minimum RE ions concentrations necessary to achieve a significant amplification can be estimated as a function of . For instance, if , amplification becomes significant for . However, to achieve this gain, a high doping level is mandatory, and , approximately. Except for low values (<10 mW), the circulating pump power has a small influence on . For larger values of the requirement for high doping level is more and more demanding. Therefore, in practice, the available RE doping level limits the value of for an amplifying MRR and the range of d and the corresponding . For , the central coupling gap is and . According to Ref. , for this value of d, low additional coupling losses both for pump and signal could be feasible. Once is determined, from Fig. 4 and depending on and , the pump intensity enhancement factor, , is obtained and, subsequently, the pump power that has to be the input in the MRR. Although the circulating pump power had a small influence in , together with the RE ions concentrations, determines (see Fig. 5) and .
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 () and additional coupling losses () are considered. The net gain that can be obtained in the MRR amplifier is evaluated as:
Net gain dependence on is plotted in Fig. 8 for three values of and (a) and (b) .
As the additional losses increase, the value of (and accordingly of the RE ion concentrations) necessary to achieve positive gain becomes larger. For instance, if , then , which implies . Once positive net gain is achieved, the rate of growth is higher with lower .
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, , for different values of the additional coupling losses is plotted. The great influence on these requirements of is clearly appreciated in this figure. As an example, if , the threshold signal gain coefficient rapidly increases with and for = 0, 0.005, 0.01 and 0.015, we obtain 31.5 m-1, 83.1 m-1, 134.9 m-1 and 187.0 m-1, respectively. Hence, in order to achieve the necessary , even small defects in the couplers fabrication process could only be compensated by notably raising the RE doping level. It has to be emphasized that the unavoidable requirements of high RE concentrations impose a host material with a high solubility for RE ions, as phosphate glass where high dopant concentration can be achieved without serious ion clustering .
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, , that is defined as . We limit the relative variation between -0.2 and, for simplicity 0.2, we assume the same additional coupling losses for both couplers. A particular attention is going to be paid to active critically-coupled structures and to compare their performance to the passive ones.
5.1. Asymmetry influence on pump enhancement
Pump enhancement presents a maximum as a function of for each in a symmetric structure for a given value of . This maximum shifts towards higher values and rapidly decreases as additional losses increase .
In Fig. 10, the evolution of the maxima position and value are represented as a function of for different values of . It is clear from Fig. 6 that (maximum value shifts towards lower ) favours pump enhancement (in the limited range of values achievable for in Fig. 2). The effect of the maximum value reduction is attenuated by the saturation of small signal gain coefficient even for low circulation pump power in Fig. 3(b).
5.2. Asymmetry influence on the drop/input port intensity rate, I41
As we did with symmetric structures, net gain for asymmetric MMR is calculated. In Fig. 11, the evolution with of the dependence of net gain with for and is plotted. Although the minimum value of does not change, the rate of growth is larger for .
Next, we study the performance in CC conditions. Differently from the passive MRR, in an active structure the value of that cancels the throughout intensity depends on the additional losses and on the signal gain amplitude coefficient for a given .
In Fig. 12, the values of for CC are plotted as a function of for four values of and for (a) and (b) . Unlike the passive structure, performance output in the drop port is not maximized for CC. The net gain that can be obtained with the parameters used in Fig. 11 is plotted in Figs. 13(a) and 13(b). Although lower net gain can be attained compared with other asymmetric configurations (see Fig. 11), significant net gain can still be achieved in case the through contribution has to be minimized.
5.3. Asymmetry influence on threshold gain
Finally, changes in are analysed when asymmetric configurations are considered. Values of are plotted as a function of for different combinations of in Fig. 14.
In Fig. 14 we can see how the necessary threshold gain value decreases for . This reduction is more significant for the higher additional coupling losses and contributes to relax the requirement for very high dopant concentrations.
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.
This work was partially supported by the Spanish Ministry of Economy and Competitiveness under the FIS2010-20821 and TEC2013-46643-C2-2-R projects, by the Diputación General de Aragón, el Fondo Social Europeo and by a grant of the Romanian National Authority for Scientific Research, CNDIUEFISCDI, project number PN-II-PT-PCCA-2011-71 "Integrated Smart Sensor System for Monitoring of Strategic Hydrotechnical Structures HydroSens".