Open access peer-reviewed chapter

Optimized Design of Yb3+/Er3+-Codoped Phosphate Microring Resonator Amplifiers

By Juan A. Vallés and R. Gălătuş

Submitted: April 16th 2015Reviewed: October 16th 2015Published: December 16th 2015

DOI: 10.5772/61767

Downloaded: 967

Abstract

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.

Keywords

  • Active integrated microring resonators
  • Yb3+/Er3+-codoped glass
  • energy-transfer inter-atomic mechanisms
  • gain/ oscillation requirements
  • asymmetric structures

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 Yb3+/Er3+-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 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 [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 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 [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 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.

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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 βc=βjα+jg. In this expression, βis the phase propagation constant, αis the loss coefficient (due to scattering and bend) and gis the gain coefficient. This coefficient describes the evolution of the pump/signal mode amplitudes caused by their interaction with the RE ions.

Figure 1.

A microring resonator side-coupled to two parallel straight waveguides for pump and signal input/output. The scheme is not to scale.

In Fig. 1 r is the microring radius and the central coupling gaps between each waveguide and the ring are di(i=1,2). Lossless intensity coupling and transmission coefficients at coupler ciare Κi0and Ti0, satisfying Ki0+Ti0=1. Correspondingly, κi0=(Κi0)1/2and ti0=(Ti0)1/2are 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 [20]. Γidenotes the coefficient for additional intensity loss at the ithcoupler. Therefore, the actual intensity coupling and transmission coefficients are Ti=(1Γi)Ti0, Ki=(1Γi)Ki0, which verify the relation Ti+Ki=(1Γi), whereas ti=Ti1/2and κi=Ki1/2are the amplitude coupling and transmission coefficients, respectively. Mode confinement guarantees that interaction between the microring and bus waveguide cores is negligible outside the coupler regions.

Amplitudes at the couplers output ports.If the input/output complex amplitudes at the couplers ports are denoted as aiand bi(i=1,2for the input/through ports at coupler 1 and i=3,4for 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:

Coupler I:|b1b2|=|t1jκ1jκ1t1||a1a2|;coupler II:|b3b4|=|t2jκ2jκ2t2||a3a4|E1

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

b1=t1a1jκ1a2E2
b2=jκ1a1+t1a2E3
b3=t2a3jκ2a4E4
b4=jκ2a3+t2a3E5

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

a2=b3exp(jφ)E6
a3=b2exp(jφ),E7

where φ=πrβc. 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:

b1=t1(1Γ1)t2exp(j2φ)1t1t2exp(j2φ)a1E8
b2=jκ11t1t2exp(j2φ)a1E9
b3=jκ1t2exp(jφ)1t1t2exp(j2φ)a1E10
b4=κ1κ2exp(jφ)1t1t2exp(j2φ)a1E11

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:

I11=|b1a1|2=t12+(1Γ1)2t22δ22(1Γ1)t1t2δcos(βL)1+t12t22δ22t1t2δcos(βL)E12
I21=|b2a1|2=κ121+t12t22δ22t1t2δcos(βL)E13
I31=|b3a1|2=κ12t22δ1+t12t22δ22t1t2δcos(βL)E14
I41=|b4a1|2=κ12κ22δ1+t12t22δ22t1t2δcos(βL)E15

where L=2πris the length of the ring and δ=exp[(gα)L]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 Γ1=Γ2=0, and there is no ring roundtrip loss, δ=1, we obtain I11+I41=1. Moreover, if in Eq. (12) κ2=Γ2=0, and hence t2=1, we obtain the through intensity rate of an all-pass ring resonator with only one coupler:

|b1a1|2=t12+(1Γ1)2δ22(1Γ1)t1δcos(βL)1+t12δ22t1δcos(βL)E16

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 (βL=2mπ, 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.

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 rate15:

E=1πR0πR(I21+I31)exp[2(gα)]dz==1πR0πRκ12{1+t22δ}(1t1t2δ)2exp[2(gα)]dz==κ12{1+t22δ}(1t1t2δ)2{1δ}(αg)LE17

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

I11={t1(1Γ1)t2δ1t1t2δ}2,I41={κ1κ21t1t2δ}2δE18

The intensity rate to the through port, I11, cancels when the critical coupling (CC) condition is verified:

t1=(1Γ1)t2δE19

Fulfillment of the CC condition produces the complete destructive interference between the internal field coupled into the output waveguide and the transmitted field in c1and, as a consequence, the transmitted intensity drops to zero. From Eq. (18), it can be concluded that if gα>0(i.e. δ>1),intensity rates I11and I41may 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, I11and I41tend to infinity and the oscillation condition is reached. The threshold gain coefficient, gth, can be calculated as:

gth=αsln[1t1t2]2πrE20

Finally, if g>gth, 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 (4I15/24I11/2) and the Yb3+-ion emission spectrum (2F5/22F7/2), 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 [25].

Figure 2.

Energy level scheme of the Yb3+-Er3+-codoped system

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. [16]. In this model the temporal evolution of the population densities of the levels, ni(i=1,5), is described by the rate equations for the Yb3+/Er3+-codoped system, which can be written as follows:

dn2dt=W12n1+CBTn1n5[A2+W21]n2CET(nYb)n2n3E21
dn4dt=W34n3[A4+W43]n42CUP(n4)n42+A5n5E22
dn5dt=CBTn1n5+CET(nYb)n2n3+W35n3+CUP(n4)n42[A5+W53]n5E23
n1+n2=nYbE24
n3+n4+n5=nErE25

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 n1(x,y,z), n2(x,y,z), n3(x,y,z), n4(x,y,z)and n5(x,y,z), respectively. Notice that, for the sake of simplicity, in Eqs. (21)—(25), the spatial dependence (x,y,z)of the population densities and the densities of stimulated radiative transition rates is omitted. Furthermore, nYband nErdenote the homogeneous ytterbium and erbium ions concentrations. In Eqs. (21)—(25), Airepresents the spontaneous relaxation rate from level i, whereas the values of the densities of stimulated radiative transition rates, Wij(x,y,z), can be obtained using the equation

Wij(x,y,z)=νσij(ν)hνΨ(x,y,ν)×P(z,ν)E26

where Ψ(x,y,ν)is the normalized mode envelope [26] of the pump, signal or co- and counter-propagating amplified spontaneous emission (ASE±) waves, with optical frequency ν. Ψ(x,y,ν)is assumed to be z-independent and depends on the index profile and waveguide profile and the waveguide geometry. Besides, in Eq. (26), P(z,ν)are the z-propagating total optical powers and σij(ν)are the absorption/emission cross sections corresponding to the transition between the ithand jthlevels. Concerning the energy-transfer inter-atomic mechanisms, a term proportional to the Er3+-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, CUP(n4). Finally, the Yb3+⇒ Er3+ energy transfer and back transfer coefficients are CET(nYb)and CBT, respectively.

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 [27], that of the Er3+-ion levels 4I13/2and 4I11/2are 7.9 ms[28] and 3.6 x 105 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 n4(x,y,z)and saturates at the kinetic limit in the case of infinite pump power [31]. 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 [32]. 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, CBT=1.5x1022m3/s[33].

ParameterSymbolValue
Signal wavelengthλs1534 nm
Pump wavelengthλp976 nm
Decay rate of Yb3+2 F5/2A2909 s-1
Decay rate of Er3+4 I13/2A4127 s-1
Decay rate of Er3+4 I11/2A53.6 x 105 s-1
Absorption cross section Yb3+2F7/2 at λpσ1210.9 x 10-25 m2
Emission cross section Yb3+2F5/2 at λpσ2111.6 x 10-25 m2
Absorption cross section Er3+4I15/2 at λpσ351.5 x 10-25 m2
Emission cross section Er3+4I11/2 at λpσ539.6 x 10-26 m2
Absorption cross section Er3+4I15/2 at λsσ345.4 x 10-25 m2
Emission cross section Er3+4I13/2 at λsσ435.3 x 10-25 m2
Energy transfer rate Er3+⇒ Yb3+
(4I11/2 + 2F7/24I15/2 + 2F5/2 )
CBT1.5 x 10-22 m3/s
Upconversion critical radiusRu9.95 Å
Ratio between critical radiiRm/Ru601/6

Table 1.

Parameters used for the gain calculations

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:

dPp(z,νp)dz=σ53(νp)N5(z,νp)σ35(νp)N3(z,νp)+σ21(νp)N2(z,νp)σ12(νp)N1(z,νp)α(νp)E27
dPs(z,νs)dz=σ43(νs)N4(z,νs)σ34(νs)N3(z,νs)α(νs)E28

In Eqs. (27) and (28), Pγ±(z,νγ)are the optical powers, where z is the distance along the waveguide axis and the label γis pfor pumping and ssfor signal. The pump and the signal are assumed to be monochromatic and the wavelength-dependent scattering losses are denoted as α(νγ)and their λ dependence is assumed to follow Rayleigh λ-4 law. Finally, in Eqs. (27)—(28) the coupling parameters, Ni(z,νs), are the overlapping integrals between the normalized intensity modal and the ithlevel population density distributions over A, which is the active area,

Ni(z,ν)=AΨ(x,y,ν)ni(x,y,z)dxdyE29

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.

ParameterValue
Waveguide cross section1.5 µm x 1.5 µm
Substrate refractive index1.51
Core refractive index1.65
Pump wavelength976 nm
Signal wavelength1534 nm
Pump mode confinement factor0.962
Signal mode confinement factor0.757
Microring radius15.47 µm
Pump wavelength resonant order156
Signal wavelength resonant order96
Propagation loss amplitude coefficient0.25 dB/cm

Table 2.

Passive parameters of the structure

The amplitude coupling ratios for pump and signal at each coupler are functions of di(i=1,2). In Fig. 3, we plot the ratios evaluated according to Ref. [35]. Particularly, for the more confined pump power a limited range of values is available.

Figure 3.

Pump (λ=976nm) and signal (λ=1534nm) amplitude coupling ratios as a function ofdi(i=1,2)[21].

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, κp0, and the additional pump coupling loss, Γp, the pump intensity enhancement factor, Ep, is basically determined by the pump amplitude gain coefficient, gp, which reflects the attenuation induced on the pump power by the stimulated transitions in the RE ions. In Fig. 4, Epis plotted as a function of κp0for 5 values of gpwith Γp=0.005. As the pump is more attenuated (the absolute value of gpincreases), the maximum Epdiminishes and is obtained for larger κp0.

Figure 4.

Pump enhancement factor as a function of the lossless amplitude coupling coefficient,κp0, for different values of the pump gain coefficient [20].

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 (nYb,nEr)where concentration units are 1 x 1026 ions/m3. RE ions concentration values were chosen with nYb= 2nEr, 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.

Figure 5.

Pump amplitude gain coefficient in a lossless waveguide as a function of the average circulating pump power for five concentration pairs(nYb,nEr). Units for the RE concentrations are 1 x 1026 ions/m3 [20].

As shown in Fig. 4 and 5, gpdepends on the circulating pump power but, in its turn, Epis a function of gp. In practice, for given concentration values, if the required average circulating pump to achieve a signal gain coefficient value is calculated, the associated gpcan 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, I41, on the lossless coupling and on the signal gain coefficient. In Fig. 6, I41is plotted as a function of the signal gain coefficient for four values of the lossless amplitude coupling coefficient and Γp=0.005.

Figure 6.

Intensity rate between the drop and the input ports as a function of the signal gain coefficient for four values of the lossless amplitude coupling coefficientκs0[20].

For each value of κs0, I41does not grow significantly until gsapproaches the threshold gain (when I41tends to infinity). Then, the input signal is strongly amplified and the rate of growth of I41is higher for lower κs0. Over the gain threshold laser operation is achieved. As we did with gp, we now calculate gsas a function of the circulating pump power for five pairs of dopant concentrations.

Figure 7.

Signal amplitude gain coefficient in a lossless waveguide as a function of the average circulating pump power for 5 concentration pairs(nYb,nEr). Units for the RE concentrations are 1 x 1026 ions/m3 [20].

As it can be appreciated in Fig. 7, gssaturates 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 κs0. For instance, if κs0=0.05, amplification becomes significant for gs80m1. However, to achieve this gain, a high doping level is mandatory, nYb=10x1026ions/m3and nEr=5x1026ions/m3, approximately. Except for low values (<10 mW), the circulating pump power has a small influence on gs. For larger values of κs0the requirement for high doping level is more and more demanding. Therefore, in practice, the available RE doping level limits the value of κs0for an amplifying MRR and the range of d and the corresponding κp0. For κs0=0.05, the central coupling gap is d0.4μmand κp00.006. According to Ref. [36], for this value of d, low additional coupling losses both for pump and signal could be feasible. Once κp0is determined, from Fig. 4 and depending on Γpand gp, the pump intensity enhancement factor, Ep, 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 gs, together with the RE ions concentrations, determines gp(see Fig. 5) and Ep.

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 (κλ0=κ1,λ0=κ2,λ0) and additional coupling losses (Γλ=Γ1,λ=Γ2,λ) are considered. The net gain that can be obtained in the MRR amplifier is evaluated as:

NetGain(dB)=10log(I41)E30

Net gain dependence on gsis plotted in Fig. 8 for three values of κs0and (a) Γs=0.005and (b) Γs=0.01.

Figure 8.

Net gain as a function ofgsfor three values ofκs0, for (a)Γs=0.005y (b)Γs=0.01[21].

As the additional losses increase, the value of gs(and accordingly of the RE ion concentrations) necessary to achieve positive gain becomes larger. For instance, if Γs=0.01, then gs>100m1, which implies nEr>7x1026m3. Once positive net gain is achieved, the rate of growth is higher with lower κs0.

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, κs0, for different values of the additional coupling losses is plotted. The great influence on these requirements of Γsis clearly appreciated in this figure. As an example, if κs0=0.05, the threshold signal gain coefficient rapidly increases with Γsand for Γs= 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 gth, 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 [25].

Figure 9.

Threshold signal gain coefficient,gth, as a function of the lossless amplitude coupling coefficient,κs0, for four different values of the additional coupling lossesΓs[21].

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, Δκr,λ0, that is defined as Δκr,λ0=(κ2,λ0κ1,λ0)/κ1,λ0. 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 κp0for each Γpin a symmetric structure for a given value of gp. This maximum shifts towards higher κ1,p0values and rapidly decreases as additional losses increase [20].

Figure 10.

Evolution of the position and value of the pump enhancement maxima as a function ofΔκr,p0for different values ofΓ1,s: (a)κ1,p0and (b)Ep[21].

In Fig. 10, the evolution of the maxima position and value are represented as a function of Δκr,p0for different values of Γ1,s. It is clear from Fig. 6 that Δκr,p0>0(maximum value shifts towards lower κp0) favours pump enhancement (in the limited range of values achievable for κp0in 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 Δκr,s0of the dependence of net gain with gsfor κ1,s0=0.1and Γ1,s=0.005is plotted. Although the minimum value of gsdoes not change, the rate of growth is larger for Δκr,s0<0.

Figure 11.

Evolution withΔκr,s0of the dependence of net gain withgsforκ1,s0=0.1andΓ1,s=0.005[21].

Next, we study the performance in CC conditions. Differently from the passive MRR, in an active structure the value of κ2,s0that cancels the throughout intensity depends on the additional losses and on the signal gain amplitude coefficient for a given κ1,s0.

Figure 12.

Lossless signal amplitude coupling coefficientκ2,s0for CC as a function ofgsfor (a)Γ1,s=0.005and (b)Γ1,s=0.01, for four values ofκ1,s0[21].

In Fig. 12, the values of κ2,s0for CC are plotted as a function of gsfor four values of κ1,s0and for (a) Γ1,s=0.005and (b) Γ1,s=0.01. 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.

Figure 13.

Net gain obtainable as a function ofgsfor the asymmetric CC configurations considered inFig.12for (a)Γ1,s=0.005and (b)Γ1,s=0.01, for four values ofκ1,s0[21].

5.3. Asymmetry influence on threshold gain

Finally, changes in gthare analysed when asymmetric configurations are considered. Values of gthare plotted as a function of Δκr,s0for different combinations of (κ1,s0,Γ1,s)in Fig. 14.

Figure 14.

Variations in the threshold signal gain coefficient,gth, as a function ofΔκr,s0for different combinations of(κ1,s0,Γ1,s)[21].

In Fig. 14 we can see how the necessary threshold gain value decreases for Δκr,s0<0. This reduction is more significant for the higher additional coupling losses and contributes to relax the requirement for very high dopant concentrations.

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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.

Acknowledgments

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".

© 2015 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Juan A. Vallés and R. Gălătuş (December 16th 2015). Optimized Design of Yb3+/Er3+-Codoped Phosphate Microring Resonator Amplifiers, Some Advanced Functionalities of Optical Amplifiers, Sisir Kumar Garai, IntechOpen, DOI: 10.5772/61767. Available from:

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