Spectral behavior of high-power distributed feedback lasers

The mode hopping behavior of high-power distributed feedback lasers emitting near 780nm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$780\,\hbox {nm}$$\end{document} is studied. The lasers have highly reflective rear and anti-reflection coated front facets. The influence of the phase of the grating at the rear facet is investigated by means of numerical simulations in the time domain taking into account spatial hole burning and self-heating effects. The simulation results are in agreement with experimental findings.


Introduction
Compact, efficient and robust tunable single frequency diode lasers emitting with narrow linewidth are key components for a broad range of sensing and spectroscopy applications (Nasim and Jamil 2013). Especially quantum technology applications need wavelengths accessible by GaAs-based devices which can directly cover the wavelength range from about 600 to 1200 nm , e.g. optical transitions in 87 Rb (McGilligan et al. 2022). Integrated devices with internal frequency selective elements are ideally suited for this purpose. Ridge waveguide distributed Bragg reflector (DBR) and distributed feedback (DFB) lasers are able to match those requirements. However, DBR lasers suffer from mode hops when operation conditions like the voltage applied to the devices or the device temperature are changed due to a thermal detuning between gain and grating sections (Radziunas et al. 2011). In contrast, properly designed phase shifted DFB lasers where both facets are anti-reflection (AR) coated can be continuously tuned over a wide range of operating parameters . DFB lasers with the front facet being AR coated and the back facet being highly reflective (HR) offer the advantage of a higher optical output power at the front facet and most often a well-behaved wavelength tuning behavior. Their spectral properties are dependent on the position of the facet with respect to the Bragg grating (Matsuoka et al. 1985). To this day it is an unsolved technological challenge to efficiently control the facet position with respect to the Bragg grating of those structures during processing. This leads to DFB lasers with gratings incorporating an uncontrolled phase at the rear facet. Mode hops, i.e. noncontinuous wavelength changes under continuously altered operation conditions like the applied current may occur for certain phase conditions. Treatments of the laser properties at threshold with respect to the variation of the Bragg grating phase at the facets can be found e.g. in (Wenzel et al. 2004). However, there is no thorough description for the mode hop phenomenon above threshold for those structures to the authors knowledge. In this work the spectral behavior of such HR-AR coated DFB lasers is analyzed above threshold by time-dependent traveling-wave-based simulations which take spatially varying carrier densities as well as self-heating self-consistently into account. The aim is to get a better understanding of the underlying processes which lead to the occurrence of mode hops and to identify device parameter ranges for stable operation.

Device structure
The DFB lasers analyzed have uniform gratings along the entire resonator. The real valued coupling coefficient of the grating is = 4.5 ∕ cm and a resonator length of L = 1500 μm is chosen to achieve a rather uniform longitudinal intensity distribution along the resonator. The ridge width is x r = 2.2 μm . The rear power reflectivity R rear is set to 0.96. The phase of the Bragg grating at the rear facet at z = L is accounted for by introducing the complex amplitude reflection coefficient r rear = √ R rear e −2 i rear . The rear facet phase can be described by rear = NΔl rear ∕Λ where Δl rear is the position of the facet as shown in Fig. 1, Λ the period and N the order of the grating. The actual cavity length is L − Δl front + Δl rear . A front facet phase can be introduced accordingly. However, the power reflectivity of real devices that were considered is typically in the order of R front ≤ 10 −3 . Thus, a variation of the phase of the Bragg grating at the front facet has a lower impact on the behavior compared to the back facet. Therefore the front facet power reflectivity is set to 0 in the simulations in order to exclude any perturbations from additional phase variations of the grating at the front facet. The Bragg wavelength is set to 0 = 783 nm . The basic design of the laser structure is described in (Wenzel et al. 2004).

Simulations with time-dependent traveling-wave equations
The main simulations were done using the BALaser software kit originally written to simulate the dynamics in edge emitting broad-area lasers (Radziunas 2022). Its optical model is based on time-dependent traveling-wave equations with slowly varying amplitudes for the forward and backward propagating complex optical fields u ± (x, z, t) where x and z describe the lateral and longitudinal coordinates with n g = 3.9 being the group refractive index, c the speed of light, n = 3.3 the reference background index, D the dispersion operator, f ± sp a term describing the spontaneous emission and k 0 = 2 ∕ 0 . The propagation constant Δ depends on the carrier density N, temperature T as well as the photon density ‖u‖ 2 = �u + � 2 + �u − � 2 and is defined by Here, Δn 0 is the difference of the effective index and the reference background index. The dependencies on the carrier density are accounted for by using . The transparency carrier density is N tr = 1.5 × 10 24 ∕ m 3 , the gain prefactor is g � = 44.8 ∕ cm and the gain compression factor is g = 1.74 × 10 −25 m 3 . The losses are described by = 0 + f N N with the background absorption 0 = 1.45 ∕ cm and the free carrier absorption cross-section f N = 1.9 × 10 −23 m 2 . The temperature dependent index shift is defined by Δn T = ∫ n � T (T − T hs )| | 2 dy with T being the temperature, T hs the temperature at the heat sink, n � T = 2.5 × 10 −4 ∕ K and | (y)| 2 the mode profile ). Equation (1) is solved with respect to the boundary conditions u + (z = 0) = 0 and u − (z = L) = r rear u + (z = L) with BALaser rear = − rear being the phase defined in BALaser (Radziunas and Čiegis 2014). Additionally, BALaser accounts for the spatially and temporally varying carrier density which is described by a diffusion equation coupled to the injection current where D eff is the effective diffusion coefficient, j the injected current density in the active region, e the elementary charge, d = 14 nm the quantum well thickness and R the recombination rate. The injected current density is described by Here, y is the derivative in vertical direction, p the electrical conductivity of the holes and p their quasi-Fermi potential which is described by a Laplace equation ∇ p ∇ p = 0 in the p-doped region to include current spreading. The quasi-Fermi potential is set to be equal to the applied voltage at the contact stripe p = U 0 and the Fermi voltage in the active region at its interface p = F . For all other boundaries the normal derivative of the quasi-Fermi potential n p = 0 . The effective diffusion coefficient is described by D eff = p N F ∕e and the electrical conductivity by p = e p (p 0 + N) with the hole mobility in the active region p = 1.71 × 10 −2 m 2 ∕ (V s) and the equilibrium hole carrier density p 0 = 1.22 × 10 −2 ∕ m 3 (Zeghuzi et al. 2018). Additionally, self-heating is taken into account by modeling the thermal dependence by a stationary two dimensional heat flow equation neglecting longitudinal heat flow Radziunas et al 2019) with the heat capacity c h and the heat conductivity L taken from (Piprek 2003). The heat source h is composed of Joule heating as well as absorption, recombination and quantum defect heat sources and solved with respect to the boundary conditions L n T = (T hs − T)∕r th at the heat sink and n T = 0 at all other outer bounds. The heat sink temperature is T hs = 300 K and r th is the inverse heat transfer coefficient which was set to match experimental continuous wavelength tuning data. The heat sources are provided by the electro-optical model, which itself is affected by the temperature distribution obtained from the thermal model. Thus, the solution is refined in an iterative procedure. The presented simulation results are calculated from the third iteration step. The total time interval of the simulations was set to 10 ns and the last 3 ns were used for calculating the results. The spatial resolution of the simulated devices was Δx = 0.1 μm and Δz = 5 μm which results in a time step Δt = Δzn g ∕c = 0.065 ps . For the calculations rear was varied from 0 to 2 in steps of 0.1 × 2 with applied voltages ranging from 1.7 to 2 V in steps of 0.05 V . Those voltages correspond to operating currents from well above threshold to slightly above 200 mA which is a typical range applied to real devices.

Simulations with stationary coupled-wave equations
For comparison the devices were additionally simulated at threshold using counter propagating stationary coupled waves a ± (z) which obey with the complex propagation factor and Δn being the difference between the effective index of refraction and the reference index. The modal gain g is treated as a parameter and a 0 is the absorption coefficient (Dumanov et al. 2015;Tawfieq et al. 2017). The complex frequency Ω is the eigenvalue of Eq. (6) which can be solved with the boundary conditions a − (L) = r rear a + (L) and a + (0) = 0 . The threshold gain g th is obtained by solving ℑ Ω g = g th = 0 , the wavelength of the supported cavity modes is given by = 0 + (d ∕d )ℜ(Ω) and the intensity within the cavity scales with |a + | 2 + |a − | 2 . The modal absorption was set to a 0 = 2 ∕ cm for these simulations.

Results
The simulation results above threshold obtained with BALaser are presented in Fig. 2. Panel (a) exemplifies the spectra as a function of the applied current for the device with rear = 0.5 × 2 . One can clearly identify a discontinuity of the emitting peak wavelength as function of the applied current at about 150 mA . The other small jumps in wavelength are caused by the discrete spacing of the injected currents in the simulations. The peak wavelength obtained from these spectra are marked to distinguish easily between the continuous tuning of the wavelength and the mode hop. The extracted peak wavelength of all simulated spectra are plotted in panel (b). Mode hops can be observed for rear facet phases of 0.3 × 2 , 0.4 × 2 and 0.5 × 2 within the chosen operating range. At 0.3 × 2 Fig. 2 Panel (a) shows complete spectral data for the simulated operating range for a selected Bragg grating phase at the rear facet with a mode hop at about 150 mA . The other small jumps in wavelength are caused by the discrete applied current spacing of the simulations. As a guide to the eye the peak wavelength tuning is indicated as dashed line. Panel (b) and (d) show peak wavelength tuning by current extracted from the spectral data. In panel (d) thermal effects due to self-heating were ignored. For comparison experimental data that incorporate a mode hop is depicted in (c) the mode hop appears at 50 mA near to laser threshold. The current where the mode hop occurs rises with increasing phase at the rear facet. The continuous shift of wavelength is about 0.7 × 10 3 nm ∕ mA when there is no mode hop present. The wavelength jump at a mode hop is 0.075 nm.
Panel (c) shows experimental peak wavelength data as a function of the injected current from a fabricated laser similar to the simulated one. The experimental findings reproduce the simulation results, namely the occurrence of a mode hop of 0.075 nm to a longer wavelength. The continuous wavelength shift is 1.6 × 10 3 nm ∕ mA and is mainly caused by self-heating when compared to simulations with self-heating turned off in panel (d). In this case, similar mode hops occur as well for a smaller range of phase conditions at the rear facet.
A more detailed analysis of the simulated time-averaged field intensity and carrier density distribution of the devices with a mode hop reveals actually a stronger local decrease of the optical field intensity distribution at the back facet compared to the other devices. This behavior gets even more pronounced with higher injected currents and near to a mode hop as shown in Fig. 3a and b. This is consistent with a carrier accumulation depicted in Fig. 3c and d. The observed local decrease of the field intensity distribution for the devices is on the one hand caused by the expected asymmetric distribution in HR-AR coated diode lasers which is accompanied by a locally smaller refractive index in comparison to the front of the cavity and a reduction of the Bragg grating phase at the rear facet. Near to a mode hop the stronger drop in the field intensity distribution is most probably caused by shifting the grating phase at the rear facet towards the condition where the reflected field from the facet does lead to out-of-phase reflections from the grating with respect to the incoming field at the Bragg wavelength. Hence there is an additional self-induced reduction of the refractive index and the Bragg grating phase at the rear facet. The described mechanism is consistent with the occurrence of mode hops at higher currents for devices with rear = 0.4 × 2 and 0.5 × 2 , respectively.
The comparative simulations with stationary coupled wave equations at threshold are consistent with the results from BALaser and are presented in Fig. 4a and b. The lasing mode switches at rear = 0.25 × 2 when the threshold gain difference of the two lowest threshold modes Δg reaches zero. A mode with higher wavelength starts lasing at 0.25 × 2 by lowering the phase at the rear facet. The actual wavelength difference corresponds to the stopband width of 0.079 nm . In contrast, at rear = 0.75 × 2 the threshold gain difference of the two lowest threshold modes is at its maximum and the laser should be in stable single mode operation. In this case the reflected field from the facet leads to in-phase reflections from the grating with respect to the incoming field at the Bragg wavelength. Disregarding the non-perfect total reflection at the rear facet this corresponds to the ∕4 phase shifted AR-AR coated DFB laser with L = 3 mm which is consistent with the preceding simulations where the field intensity distribution at the rear facet peaks at rear = 0.8 × 2 for the operation range simulated.
Based on the simulation results at threshold one may define parameter regimes for stable single mode operation by introducing an optical power flatness F = ∫ cavity (Î(z) − 1) 2 dz∕L ) of the longitudinal field intensity distribution I(z) to account for spatial hole burning phenomenologically. Î (z) = I(z)L∕ ∫ cavity I(z)dz is the normalized intensity distribution along the resonator. If F ≤ 0.04 as well as ΔgL ≥ 0.05 are assumed for the limits for stable single mode operation (Ishikawa et al. 1987) the amount of diodes with stable single mode emission is about 63% . This is consistent with the results from BALaser of 70% . However, the phenomenological approach predicts a symmetrical range of Bragg grating phases at the rear facet around 0.25 × 2 with no stable single mode operation. From the simulations above threshold this range is clearly shifted towards Additionally, we analyzed the light output power versus operating current (LI) characteristics of these devices in relation to the spectral characteristics and the change of the Bragg grating phase at the rear facet with BALaser. Figure 5a and b show the LI characteristics at the front and rear facet for a selection of Bragg grating phases at the rear facet. A higher output power is observed at the front facet as expected due to the asymmetric coatings and field intensity distribution within the resonator. However, the Fig. 4 Wavelength of the lasing modes of the device at threshold a as well as the relative threshold gain difference of the two lowest threshold modes (dashed line) and the optical power flatness (solid line) b as a function of the Bragg grating phase at the rear facet slope of the LI curves at the rear facet is found to be highly dependent on the Bragg grating phase at the rear facet and reaches its lowest value for devices where a mode hop is observed. The largest maximum output power ratio between different devices where rear varies between 0 × 2 and 1 × 2 is greater than 200% and is in contrast to the LI characteristics at the front facet where the deviation is less than 1% . These findings are consistent with the simulated time-averaged field intensity distribution variations. The device with rear = 0.5 × 2 shows a nonlinear LI curve at the rear facet around 150 mA which is highly damped at the front facet. At this current the laser has a mode hop as depicted in Fig. 2. This behavior was also observed experimentally. Figure 6 shows the measured LI characteristics for the laser presented above. For comparison the optical output power at the rear and front facet was normalized by its value at 80 mA . At about 55 mA the spectral map of the laser shows a mode hop which is accompanied by a nonlinearity of the LI curve at the rear facet.

Conclusions
In this work we investigated the spectral behavior of HR-AR coated ridge waveguide DFB lasers. By means of time-dependent active laser simulations we could accurately describe experimentally observed wavelengths tuning behavior in such structures which includes continuous wavelengths tuning due to self-heating as well as the occurrence of mode hops from the lower to the higher stopband mode above threshold for a range of Bragg grating phases at the rear facet. The field intensity distribution within the resonator was found to be highly dependent on the Bragg grating phase at the rear facet which itself was shown to be affected by the self-induced local carrier accumulation at the rear facet above threshold. This mechanism can describe the occurrence of mode hops in high-power DFB Lasers with asymmetrically coated facets.