Simulation of an erbium-doped chalcogenide micro-disk mid-infrared laser source

The feasibility of mid-infrared (MIR) lasing in erbium-doped gallium lanthanum sulfide (GLS) micro-disks was examined. Lasing condition at 4.5 μm signal using 800 nm pump source was simulated using rate equations, mode propagation and transfer matrix formulation. Cavity quality (Q) factors of 1.48 × 10 4 and 1.53 × 10 6 were assumed at the pump and signal wavelengths, respectively, based on state-of-the-art chalcogenide micro-disk resonator parameters. With an 80 μm disk diameter and an active erbium concentration of 2.8 × 10 20 cm 3 , lasing was shown to be possible with a maximum slope efficiency of 1.26 × 10 4 and associated pump threshold of 0.5 mW. ©2011 Optical Society of America OCIS codes: (140.3070) Infrared and far-infrared lasers; (140.3500) Lasers, erbium; (140.3580) Lasers, solid-state; (140.3945) Microcavities; (140.5680) Rare earth and transition metal solidstate lasers. References and links 1. B. J. 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Introduction
Chalcogenide glasses (ChGs) are distinguished for having chemical durability, photosensitivity, high refractive index, low phonon energy, low melting temperature, and broad infrared transparency [1][2][3].Such characteristics make this family of glass attractive for the development of infrared integrated optical devices [4][5][6].Integration of multiple monolithic components on a single substrate is beneficial for minimization of size and cost by enabling systems-on-chip applications.A key enabler for such systems is the demonstration of monolithic light sources emitting in various wavelength regimes.
Rare earth (RE) elements are incorporated as active emission centers in passive crystalline and amorphous materials.Many RE transitions are generally quenched in hosts such as phosphate and silica glasses which have high phonon energies that can bridge low energy gaps and cause large multi-phonon relaxation rates [7,8].On the other hand, ChGs have relatively low phonon energies which reduce the possibility of these non-radiative relaxations and enable emission of long wavelengths.
In this paper, we present our simulation results toward developing a monolithic MIR laser source, utilizing erbium-doped GLS glass.Bulk erbium-doped GLS showed MIR photoluminescence emission at 4.5 µm through the transition between 4 I 9/2 and 4 I 11/2 energy levels [25].Compared to other ChG glasses, GLS is capable of hosting relatively high erbium concentrations (2.8 × 10 20 ions/cm 3 ) without being affected by luminescence quenching [25].Nevertheless, the considered transition is characterized by a small emission cross section of 2.5 × 10 21 cm 2 .This limits the maximum possible gain to less than 4 dB/cm.For lasing to be possible under this gain limitation, resonators with minimum quality (Q) factors of 3.5 × 10 4 are required.
Recently, lift-off and thermal reflow process has been used to demonstrate ChG microdisks with Q factors in excess of 10 5 at 1.55 μm [26].This is a catalyst for fabricating monolithic laser sources given the aforementioned specification requirements.In the subsequent text, lasing at 4.5 µm is examined with 800 nm pumping for erbium-doped GLS micro-disk.The rate equations of erbium, the pump, and signal disk modes were solved, and a transfer matrix formulation was used to estimate the output lasing power.

Simulation model
The developed model considers pump and signal modes that correspond to wavelengths of 800 nm and 4.5 µm, respectively.Separate bus waveguides introduce and collect the pump and signal light from the disk as illustrated in Fig. 1.Also, the following constants and assumptions were used: 1) uniform erbium doping concentration of 2.8 × 10 20 cm 3 ; 2) disk structure of 80 µm diameter and 600 nm thickness; 3) refractive indices of 2.42 and 2.35 at the pump and signal wavelengths, respectively, were obtained by fitting experimental data to a Cauchy relation [27]; 4) transverse electric (TE) polarization modes, with dominant electric component parallel to the disk plane, were included; 5) Purcell cavity enhancement is neglected since the photon density in the cavity is high and the micro-disk structure has high order modes with large radiation; and 6) unidirectional mode propagation was considered.  1. Laser configuration consists of a micro-disk with input pump waveguide and output signal waveguide.P is the pump power and S is the signal power at the positions indicated by the subscripts, κ 2 is the power coupling coefficient between the bus waveguides and the disk, and the subscripts P and S stand for the pump and signal, respectively.
We establish some useful considerations, to ease the development of our model.Since the disk has a relatively large diameter, the free spectral range will be less than the emission and absorption line-widths of erbium.Hence, resonance can be assumed for the pump and signal modes.Also, for the continuous wave (CW) case, steady state condition is imposed.With these assumptions and no input signal, the internal power in the disk can be related to the power in the pump and signal buses through the following relations [28]: .
where 2 P a is the round trip power absorption for the pump mode, P i is the pump power and S i is the signal power at the positions indicated by the subscripts, 2 , PS  is the power coupling coefficient between the bus waveguides and the disk and the subscripts P and S stands for the pump and signal, respectively.We can also eliminate the inter-effect of the signal and pump bus waveguides on the disk pump and signal modes, respectively, by recognizing the following.The signal mode has much longer wavelength than the pump mode.Therefore, it possesses a greater evanescent tail length.Hence, the signal bus can be placed far from the pump mode tail which prevents pump out-coupling.In addition, since the pump light has a short wavelength, the pump bus can be designed with a width smaller than the signal cutoff.Hence, the pump bus will also have no coupling with the signal mode.Then, P 3 and S 1 will be equal to P 2 and S 4 , respectively.P 4 and S 2 are related to P 1 and S 3 by Beer-Lambert law and the signal coupling coefficient according to the following relations [28,29]: .
where l is the azimuthal coordinate along the disk circumference, / () gl are the absorption and gain coefficients of the pump (P) and signal (S) modes at l. From Eq. ( 4) and Eq. ( 5): ).
this equation defines the condition that should be satisfied for the steady state (CW) lasing case.The out coupled signal power should be exactly recovered by the round trip gain.For smaller gain, lasing is not possible, while higher gain values do not satisfy the steady state condition.
The energy evolution of erbium is described using the five-level model as elaborated in Fig. 2 [30,31].The ion-ion and ion-photon interactions with the pump and signal light are calculated according to the subsequent rate equations: where N i is the ion concentration at energy level i, I S/P is the signal or pump beam intensity, ħω S/P is the photon energy, C ij are the energy transfer coupling coefficients that quantify the ion-ion interactions, / / ea SP  is the emission (e) or absorption (a) cross section, a ij is the spontaneous emission rate from level i to level j, W i is the multi-phonon decay rate from level i to the next lower energy level, and N Total is the total erbium doping concentration.The values of the energy transfer constants are obtained experimentally [30].The energy gap law is used to evaluate the multi-phonon decay rates [32].The spontaneous emission rate is obtained using Judd-Ofelt theorem [33].Finally, McCumber and Füchtbauer-Ladenburg relations are used to calculate the emission and absorption cross sections at the pump and signal wavelengths [34,35].The values of these parameters are provided in the appendix (Table 2).Rosenbrock iterative method was used to find the steady state population distribution of the ions.The signal gain coefficient (g S ), and the pump absorption coefficient due to erbium (α P,Er ), are functions of the pump and signal intensities.These coefficients are given, per unit area, by the following equations [29]: The cavity modes were calculated for the disk cross section in Fig. 3.A CaF 2 substrate was assumed for its low absorption in the MIR regime.As moisture can be trapped in CaF 2 , a GLS thin film to coat the entire substrate was taken into account.To reduce the signal radiation losses, a large diameter of 80 microns was assumed.Reducing the thickness of the disk minimizes scattering from the side walls while it increases the signal radiation losses.Signal radiation was found to be insignificant for a disk thickness of 0.6 µm and a substrate coating layer of 0.1 µm thickness.A full-vectorial finite difference mode solver on FIMMWAVE was used to calculate the disk mode profiles at the signal and pump wavelengths [36].Having azimuthal symmetry, the two-dimensional solution was calculated for the disk cross section along the radial and planar directions.

Disk
Coating layer Substrate 0.6 µm 0.1 µm 80µm Fig. 3.The micro-disk material cross section showing a CaF2 substrate, and erbium-doped GLS coating layer and disk.A CaF2 substrate was considered for its low absorption in the MIR regime.As moisture can be trapped in CaF2, a GLS coating layer was sandwiched between the disk and the substrate (dimensions not drawn to scale for clarity).
For optical cavities, the Q factor (Q) is used to quantify the power loss relative to the stored internal energy.An equivalent absorption coefficient (α eq ) is obtained using the following equation [37]: 2 .
where n g is the group velocity of the mode, and λ is the free space wavelength.Volume current formulation was used to estimate the mode scattering losses [38].The Q factors were evaluated based on preliminary experimental roughness parameters (10 nm roughness amplitude and 150 nm correlation length) of the demonstrated high Q (due to thermal reflow) ChG micro-disk [26].Future fabrication and characterization studies will fine tune these parameters.The radiation losses were quantified using a perfectly matched layer.The bulk absorption coefficient of GLS (0.035 cm 1 at 800 nm, and 0.006 cm 1 at 4.5 µm [39]) was multiplied by the mode confinement factor to arrive at the mode absorption losses.The signal mode gain coefficient and pump mode absorption coefficient can be obtained using:   where (x,y) is a coordinate point on the disk cross section, f S/P is the signal or pump mode power profile normalized to 1 W.This detailed model was used to simulate the micro-disk laser system under consideration.The simulation results are given in the next section.

Simulation results
Including the initial transient evolution of the mode powers and ion populations would require large simulation time.For this reason, we developed a route which utilizes the previously explained model to find a self consistent steady state solution.Initially, the signal gain was calculated as a function of the signal and pump intensities using Eq. ( 7) and Eq. ( 8) (Fig. 4).Linear interpolation was used later to find the gain for the intermediate points.The data range was chosen to cover the saturation limits.This range was discretized such that the maximum interpolation error is less than 0.2%.Erbium absorption of the pump light was also calculated in the same way using Eq. ( 9).The solutions of the fundamental signal mode and the first eight pump radial modes were calculated, as shown in Fig. 5.For the different pump modes, there are several competing factors affecting the obtained signal mode gain.First, the signal mode gain can be maximized by using the pump mode for which the maximum signal intensity overlaps the area having the maximum gain, i.e. highest pump intensity.However, as shown, the gain can decrease drastically as the signal intensity increases.In addition, concentrating most of the pump power at the signal intensity peak is of no benefit in the saturation region.Using the residual power to pump larger area of the signal mode would result in higher signal mode gain.Since it is not straightforward to identify the pump mode that results in the highest signal gain, the signal gain values assoscieated with the considered pump modes should be identified and compared.The signal mode gain was calculated as a function of the internal pump (P i ) and signal (S i ) power using Eq. ( 11).The mode intensity profiles were discretized into 50 segments per micron which results in negligible error in estimating the intensity at each grid point.The gain was interpolated at each grid point using the data obtained in Fig. 4. A comparison between the obtained signal gain by exciting several pump modes is shown in Fig. 6.It is clear that pumping the first order mode does not result in the highest possible gain.Higher values can be achieved by pumping higher order modes.However, going beyond the 8th order mode (not shown) minimizes the pump signal overlap (Fig. 5) and hence minimizes the signal mode gain.The passive cavity Q factors and the equivalent absorption coefficients (Eq.( 10)) of the TE11 pump and signal modes are summarized in Table 1.As listed, the losses for both cases are dominated by scattering.This loss is much greater for the pump mode due to its relatively small wavelength.As explained in [38], the scattering losses are directly related to the mode power amplitude within the scattering volume.For the considered pump modes, the difference in this value was found very small (less than 1%).The pump modes also have ignorable absorption and radiation losses similar to TE11.Consequently, these modes have close to equal Q factors.For this reason and because the 7th order mode gives the highest possible gain, it was chosen to pump the disk.Erbium absorption for the pump mode light was quantified using Eq. ( 12) and found to be small at operational pumping power levels.The signal power in the disk was calculated as a function of the internal pump power and the signal coupling coefficient.Due to high internal pump losses, the pump power, and hence the signal gain, would show high variations along the disk circumference.Therefore, the disk was discretized azimuthally into 1µm length segments at the disk circumference.The modes were propagated through these segments, starting from P 1 and S 1 in Fig. 1 and using Eqs.( 3)- (5).The bisectional method was used to search for the signal power (S 1 ) that satisfies the steady state lasing conditions given by Eq. ( 6).The condition was tested for a maximum tolerance of 0.1%.With the calculated Q factors and the maximum achievable signal gain, the error in estimating the calculated power is not significant and therefore can be ignored.
The round trip pump power absorption, caused by erbium and the passive cavity losses, was found to be ~75%.Based on Eq. (1), a pump coupling coefficient of 0.25 would maximize the pump power accumulation in the disk (P 1 /P in ) and minimize the needed input pump power (Fig. 7(a)).This coupling value was used in Eq. ( 1) to find the input pump power (P in ) that corresponds to P 1 .Due to the high scattering losses, the maximum power accumulation is limited to 4. This could be elevated by two orders of magnitude if the scattering losses in the disk were eliminated as shown in Fig. 7  Equation (2) was used to find the signal power out-coupled from the disk to the signal bus.As mentioned previously, the laser performance is tightly related to the signal coupling coefficient.Figure 8(a) shows the signal output power of the disk for the cavity with the scattering losses taken into account.The threshold power varies directly with the signal coupling.For coupling coefficient higher than 2 × 10 3 , lasing is not possible since the signal gain is not sufficient to recover the internal and external disk losses.For low signal coupling, the lasing threshold shows saturation at a minimum of 0.2 mW.However, the slope efficiency decreases drastically by decreasing the coupling to that level.
The output power peaks at a signal coupling of 4 × 10 4 .This value of signal coupling gives an optimized performance for the micro-disk device as it maximizes the slope efficiency (1.26 × 10 4 ) with a lasing threshold of 0.5 mW.Higher coupling values result in small signal accumulation.On the other hand, decreasing coupling below that level will increase the accumulation but only a small fraction of the internal signal power couples to the output bus.
Simulated efficiency of erbium-doped ChG fiber, can achieve ~15% [24].However, fiber lasers require long lengths (tens of centimeters) and do not offer suitable solution for on-chip applications.In contrast, the predicted slope efficiency of the micro-disk is very small but it offers a compact platform for on-chip applications.The low slope efficiency for the microdisk case is caused by the high pump scattering from the sidewall roughness.This results in a small pump accumulation in addition to a relatively high lasing threshold.Progress is taking place to reduce these losses [26], for which there is a vast untapped opportunity to enhance lasing characteristics by two orders of magnitude (Fig. 8(b)).

Appendix
Fig.1.Laser configuration consists of a micro-disk with input pump waveguide and output signal waveguide.P is the pump power and S is the signal power at the positions indicated by the subscripts, κ 2 is the power coupling coefficient between the bus waveguides and the disk, and the subscripts P and S stand for the pump and signal, respectively.

Fig. 4 .
Fig. 4. Steady state signal gain as a function of the pump and signal intensity for erbium doped GLS with concentration of 2.8 × 10 20 cm 3 .

W / m 2 Fig. 5 .
Fig. 5. Intensity distribution of the signal (S) and pump (P) modes with the total mode power normalized to 1W.Polarization indicated by the subscripts.The first number is the planar index while the second is the radial index.

Fig. 6 .
Fig. 6.Signal mode gain obtained by exciting several pump modes with different radial orders.The gain is computed as a function of the internal signal (Si) and pump (Pi) modes powers. (b).

Fig. 7 .
Fig. 7. Pump power accumulation as a function of the power coupling coefficient: (a) the continuous line shows the basic case with the scattering losses included (Pump Q = 1.48 × 10 4 ) (b) the dashed line shows the case with no scattering losses taken into account (Pump Q = 4.85 × 10 6 ).
Fig.2.Erbium energy levels and ion-ion interaction parameters (Cij).800 nm pump source excites the ions from the ground state to 4 I9/2.The excited ions decay to 4 I11/2 to emit 4.5 µm signal light.