Simulation of a DBR Edge Emitting Laser with External Air Gap Tuning Mirror
Abstract
A methodology for simulating an edge emitting laser is demonstrated that incorporates a distributed Bragg Reﬂector (DBR) as one of the mirrors, with the other being a variable air gap external planar mirror that is used as a method for ﬁne tuning the lasing wavelength. This type of calculation requires non uniformity of the dielectric’s static and dynamic (pumping) properties in both the transverse and longitudinal directions, together with self consistent coupling to the external tuning cavity. A key challenge addressed is the proper treatment of the open nature of the cavity, such that the amplitudes of the outgoing waves leaving the facets are consistent with the cavity interior and facet reﬂectivity. Mode competition and hopping are natural consequences of the methodology developed. In the model developed here, the spatial hole burning effects, cross and selfsaturation effects can be included while accounting for the degree of spatial overlap of modes. We also account for diffraction losses at the output facet which can become important in the case of strong index guiding.
Key Words: Laser, DBR, External Mirror Air Gap Tuning Cavity, Effective Medium Method, ElectroOptic modulation
Introduction
Distributed Bragg reﬂectors (DBR) play an important role in optical ﬁltering and as mirrors. They are used for frequency selective mirrors to shape the spectral response of laser cavities in both vertical cavity surface emitting and edge emitting lasers. DBRs employ Bragg reﬂection to attenuate propagation of light through their grating structure, thus creating a highly reﬂective band in their transmission function. Bragg reﬂection is used to an opposite effect in distributed feedback lasers (DFB), in which the resonator modes are the Bragg waves allowed to propagate through the grating structure that extends over the entire cavity. However, the main structural aspect common to both types of lasers is the essential nonuniformity of their refractive index along the direction of mode propagation. Of critical importance to their operation and performance are also the pumping induced nonuniformity in their interior refractive index and optical intensity. An important aspect of all laser cavities is also the fact that they are open systems from which light escapes to the outside.
In our previous articles on edge emitting (EEL) and vertical (VCSELs) lasers, we exploited the cylindrical symmetry to treat nonuniform cavities, while simultaneously keeping the problem essentially 2D or 1D. We treated the cavity as an open system only in the case of VCSELs, and ignored the longitudinal nonuniformity (along the propagation direction) in the case of rectangular EELs. Here we describe a hybrid a rectangular edge emitting open laser cavity with a DBR at one end, and in which we take longitudinal nonuniformity fully into account. We go a step further and allow for an external air gap tuning cavity by including the effects of coherent longitudinal propagation within the air gap, but with beam diffraction treated approximately as an additional loss mechanism. The longitudinal mode spectrum is calculated self consistently with electrical and thermal induced variations in the cavity refractive index.
In the hybrid DBR edge emitting laser described here, the requirement to resolve the localized optical intensity patterns is fundamentally a 3D problem, since the DBR reﬂector must resolve optical standing waves along the propagation direction, and the gain cavity must resolve modal intensities transverse to this direction. However, the problem can be reduced to 2D for cavities that are structurally uniform in one of the transverse directions, with vanishing electric ﬁeld at all boundaries of the plane, which is a reasonable approximation to real systems featuring index guiding.
The open nature of the cavity is due to the partially reﬂective surfaces at both ends, so that a fraction of the ﬁeld propagating inside transmits across the facets, and radiates out to inﬁnity. Solving for modes that are consistent with this condition requires matching both the amplitude and the derivative of the ﬁeld to the outgoing plane waves in the space outside the resonator. This boundary condition, and the frequency dependence of the dielectric function inside the cavity, result in a nonlinear eigenvalue problem for its optical modes. These modes are called quasinormal modes to emphasize the fact that their associated eigenfrequencies are complex, whose imaginary part gives the ﬁnite lifetime of the ﬁeld inside the cavity. The above boundary condition and the complex dielectric function mean that the lifetime of the mode accounts for both the absorptive losses in the cavity interior and the photon escape due to the open boundary. For readers familiar with the literature in this ﬁeld, we remark that the modes found here are characterized by real valued wavevectors outside the resonator, and complex wavevectors (in general) inside. Thus, they are actually constant ﬂux states[1, 2].
In contrast to the longitudinal mode spectrum given by ω_{n} = c/(nL) for a cavity of length L and refractive index n, the quasinormal modal frequencies must be determined from solutions of the complex Helmholtz equation. A straightforward application of this strategy can result in a prohibitively slow simulation. However our code makes use of the effective medium method as in the VCSEL simulations, as well as various software level techniques, so that the simulations described below can be completed within 12 hours each.
This article is organized as follows. We begin by outlining the methodology for creating the structure and the mesh to setup the simulation. We brieﬂy recount the important theoretical steps of the calculations carried out by the new code, and discuss their scope. We then discuss the results of several illustrative simulations for a GaAs/AlGaAs DBR laser, followed by concluding remarks.
2 Methodology
2.1 Structure Creation and Meshing
In order to facilitate easy creation of the two fundamentally different regions of a DBR edge emitting laser, namely the active laser regions and the distributed Bragg reﬂector (DBR), a combination of the normal region statements coupled with automated DBR creation statements are used. The few layers constituting the active laser regions are deﬁned ﬁrst, followed by the automated creation of the DBR mirror, which overwrites the previously created active laser regions and is at right angles to them. The entire complicated material structure can therefore be created in just a few lines.
Meshing is further simpliﬁed by separation of the structure meshing from the mesh used in the optically active laser simulations. In the case of DBR laser, the laser mesh is automatically modiﬁed to include the DBR regions generated by structure meshing. The laser mesh can then be ﬁne tuned to properly include dielectric modulations within the laser cavity, independent of the DBR regions.
2.2 Theoretical Approach
Model of the Laser Cavity
In frequency space, the electric ﬁeld satisﬁes the Helmholtz equation[3],
(1)
where c is the speed of light, and n(r,ω) is the spatially and frequency dependent refractive index. We assume n(r,ω) to have a step discontinuity at the cavity facets, so that n = 1 outside and takes arbitrary values inside the cavity, while constrained by the condition of causality. With the coordinate frame shown in Figure 1, we take E(r,ω) = 0 at the cavity boundaries in the xz plane. At the rear (y = 0) and front (y = L) facets, we impose the open boundary conditions that force the ﬁeld just outside the cavity to be an outgoing plane wave,
(2)
(3)
where k_{0}(ω) = Re[ω/c]. Since the wavevector is real outside the cavity, it yields a physical plane wave in contrast to quasibound states that diverge or vanish at inﬁnity. In recent literature on open cavities, these modes have been termed constant ﬂux states[1].
The frequency ω is realvalued so far, but it is continued analytically to complex values. The Helmholtz equation (1) is an eigenvalue equation, with eigenvalues ω corresponding to frequencies of the modes supported by the cavity. Due to the frequency dependence of the refractive index, n(r,ω), and the ωdependent boundary conditions, it is a nonlinear eigenvalue problem. To solve this eigenvalue problem, we follow our previous modeling of VCSELs (see also Wenzel et al. [4]).
For a ﬁxed ω_{0}, and letting k_{0} = ω_{0}/c, (1) transforms into a linear problem for a dimensionless eigenvalue ν,
(4)
where n_{g}(r,ω) is the group refractive index,
We assume that the laser cavity is uniform and large in the z direction and identify the modes predominantly TE or TM like, which is a good approximation when the ﬁeld conﬁnement in z direction is not signiﬁcant. We write the ﬁeld amplitude in a separable form,
(5)
where h is the cavity depth in the z direction, and we have omitted the ω dependence of X and Y for clarity. The function X(x; y) depends parametrically on y, and satisﬁes a scalar 1D Helmholtz equation in variable x. It generates the transverse mode proﬁle, with eigenvalue β(y), and deﬁnes an effective medium described by two refractive indices n_{eff} (y, ω_{0}) and (nn_{g}(y, ω_{0})),
(6)
(7)
The longitudinal propagation of the mode, with proﬁle Y (y), is then a 1D propagation inside this effective medium, governed by these effective indices. We incorporate the continuity of the ﬁelds at the boundaries, and the arbitrary variation of the refractive index in the cavity, to compute X(x; y) and Y (y) to construct a transfer matrix for the cavity. We refer to this matrix as Q(ω) and its element, Q_{11}(ω), is the inverse transfer function of the cavity for outgoing modes. The transfer function then provides the mode frequencies, and its multiplication with the frequency dependent spontaneous emission rate yields the ampliﬁed spontaneous emission spectrum[5] of the resonator.
For concreteness, we mention that for a FabryPerot cavity with facet reﬂectivities equal to r_{1} and r_{2}, the modes of this transfer function correspond to the familiar condition 1 − r_{1}r_{2}e^{−2(α−g)Lei2ωnL/c}, where α and g are internal loss and gain respectively. As in the FabryPerot cavity, ﬁnite loss due to r_{1}r_{2} < 1 and α − g implies that the poles lie at complex values of ω.
Our method lends itself naturally to modify the optical properties at the output facets by accounting for the effects of transverse conﬁnement on mode reﬂectivities and coupling to external cavities, as described below. In our methodology, the rear interface can be modeled as a ﬁlm of speciﬁed power reﬂection coeffcient, and the front facet as either a ﬁlm, or an external air gap cavity.
External Air Gap Cavity
An air gap cavity can be included in our model by specifying the location of an external mirror facing the resonator output facet. The external mirror location is outside the Atlas mesh, and therefore the cavity can be modeled without extending the laser mesh. The air gap formed between the facet and the mirror extends the cavity in both the longitudinal and transverse directions. The extension in the longitudinal direction shifts the mode spectrum due to increase in the effective optical length of the cavity. The extension in the transverse direction introduces diffraction due to free space propagation within the gap. After a roundtrip in the air gap, the diffracted beam reentering the waveguide couples to all its guided, leaky, and radiation modes. Since we aim to model cavities operating (and lasing) at a single transverse mode, this coupling becomes an effective loss mechanism for the transverse mode.
At frequency ω, the modes of the air gap are superpositions of plane waves propagating back and forth between its two ends. From this ﬁeld, and the purely outgoing ﬁeld at the outer face of external mirror, we construct an effective transfer matrix for the front facet due to the presence of an external mirror and the diffraction within the air gap [6]. The interface transfer matrices are then multiplied into the cavity transfer matrix to construct an effective transfer function for the entire structure.
Selfconsistency
Simulation of lasing must always account for the drastic changes in the refractive index to account for the bias induced changes from absorption to gain over an expanding frequency range. In addition to this, thermal effects associated with the power ﬂow further modify the frequency dependence of the refractive index. From the above discussion of theory, it is clear that the mode spectrum would also shift as a result of these changes to the refractive index. Thus it is important that the simulated mode spectrum is selfconsistent with the electrical and thermal conditions of the cavity. In our methodology, we begin with modes of a cold cavity, and converge to the solutions of coupled Helmholtz, driftdiffusion, and photon rate equations. The computation uses a lookahead strategy to compute mode spectrum only when it shifts by more than the mode line width. This results in much faster simulation, with little effect on accuracy of the solutions.
Simulations
We performed simulations for the structure shown in Figure 1. The structure consists of 100 micron long and 1 micron wide AlGaAs waveguide, with a 100 nm GaAs gain region in the center. The waveguide has a GaAs/AlGaAs DBR mirror made of 40 half cycles with widths tuned to the photon energy of 1.4 eV. In the present simulations, we use a real refractive index (set IMAG.INDEX=0 on MATERIALS statement) everywhere except in the active layer. Within the active region, we use the standard gain model implemented in Atlas, with the prefactor, GAIN0 = 6000, and we broadened the gain spectrum by a Gaussian of linewidth 5 meV. We also included free carrier absorption within the gain region by using the model ƒ = ƒ_{n}n_{c} + ƒ_{p}n_{p}, where ƒ is the free carrier absorption, and n_{c} and n_{p} are the electron and hole densities. The parameters ƒ_{n} = 10^{−18}cm^{2} and ƒ_{p} =2 • 10_{−18} cm_{2} were used to phenomenologically set the carrier related absorption loss to about 20 cm−1 . We also activated frequency dependent modal reﬂectance including diffraction losses at the front facet by specifying MODAL.REFLECTANCE and LOSS.DIFFRACTION on the LASER statement. The band gap of the active medium was set to 1.39 eV.
Figure 1. Structure used in the simulations. The z direction is implicit and into the page. The 1 micron long AlGaAs waveguide is located at the left end with GaAs layer at the center, and modeled using the standard gain model. The inset shows the AlGaAs/GaAs DBR at the top end of the waveguide.

The laser model described here is activated by the parameter HYBRID on the LASER statement. We set (NMODE=1) and (MAX.LMODES=12), to force Atlas to search upto a maximum of 12 TE like modes with the fundamental transverse mode, within the energy range 1.384 eV to 1.415 eV. Atlas performs this search at all bias points, updating the transfer function magnitude Q_{11}(ω_{0})^{–1} as the refractive index changes.
In Figure 2, we show the ampliﬁed spontaneous emission spectra at a subthreshold bias of 1.4 V: the blue corresponds to the laser cavity without any external mirror, and the red to the presence of an air gap 0.6 micron wide and formed by a mirror with power reﬂectivity of 40%. The effectively larger cavity results in redshift of the spectrum. The frequency dependence of reﬂectivities, the transmission function of the external cavity, and the refractive index further modulate the spectrum. The ASE with air gap also exhibit smaller linewidths, which is due to the slightly smaller escape rate caused by a mirror of higher reﬂectance than the approximately 30% reﬂection from the waveguide face alone (see Figure 3).
Figure 2. The effect of adding a 0.6 micron air gap on the ampliﬁed spontaneous emission spectrum (both spectra taken at 1.4 V bias the operating point A in Figure 4). The redshift of the spectrum is due to a larger effective cavity. However the shift is not uniform over the entire range of energies due to the frequency dependence of dielectric and modal reﬂectivities. 
Figure 3. Power reﬂectance at the front facet with only the dielectric step (blue) and with a mirror of reﬂectivity 40% added 0.6 microns from the waveguide facet. 
We now turn to the simulation of electrical pumping of the above structure (with no air gap). Figure 4 shows the photon densities as a function of anode current for the three most populated modes. Single mode lasing is distinctly visible in this graph. Using the SPECTRUM parameter on the SAVE statements, we also saved the spectra of cavity transmission Q_{11}(ω_{0}) ^{−1}, and modal photon densities at subthreshold (1 mA and 1.2 mA denoted as A and B in 4), and above threshold current of 1.6 mA (denoted as C).
Figure 4. Modal photon densities (1/cm) of the three most populated modes as a function of anode current. The graph clearly shows single mode lasing behavior. A, B, and C mark three operating points at which we examine the mode spectra. 
Figure 5 shows the evolution of the cavity transmission function Q_{11}^{−1} as the bias current is tuned far into the lasing regime. These curves convey two important physical points. First, the peaks blue shift due to reduction in the real part of the refractive index, which in turn is the result of rising gain. This effect can be understood roughly from the FabryPerot picture where the modal frequencies are given by ωm = mL/ (n_{r}c) for integer m and real refractive index nr. Second, the slowly varying modulation of the spectra shows a remarkable change as we move above threshold. This is caused by gain compensation of the net loss due to both absorption and escape from facets. We can see from this spectrum, and conﬁrm from a plot of photon energies of the modes, that lasing occurs near the highest peaks inQ_{11}^{−1}.
Figure 5. (a) Spectrum of the transmission coeffcient, Q_{11}^{−1} taken at the three operating points indicated in Figure 4. The peaks in this spectra deﬁne the reference frequencies in Eq. 4. 
The actual modal frequencies are slightly shifted from the location of peaks in the transmission function. The location of a pole relative to the peak on the real axis is controlled by the overall phase of the function Q11(ω). In addition, by allowing the refractive index to change linearly in the complex domain, we effectively tune the optical path length of the wave, which becomes n(ω0)(1 −ν/2). This results in shift of the modal frequencies with respect to the reference frequencies identiﬁed from Figure 5.
In Figure 6, we plot the photon densities at modal energies at the three operating points shown in Figure 4. We found that the real part of the complex resonator eigenfrequencies were shifted relative to the peaks of the transmission function by about 0.04 meV. This additional shift arises from the phase accumulation within the cavity, the DBR, and the phase shifts associated with the output facet. This additional shift is often small enough to be ignored. If these shifts are significant, then this particular behavior of modes must be closely scrutinized with respect to the material models, parameters, and the geometry of the laser. We do not investigate this further here due to the scope of this introductory article.
Figure 6. Modal photon densities (1/cm) for all 12 modes in the simulations at the three operating points indicated in Fig. 4. Photon energies blue shift with increasing gain, which leads to decrease in the phase velocity index. The energy of each mode is further shifted by changes in the optical path length, proportional to Re(ν/2), resulting in a nonuniform frequency spacing of modes. The latter shift is highly sensitive to the models and material parameters used for cavity dielectric and facet reﬂectivities. 
Conclusion
In summary, we have demonstrated a methodology for simulating rectangular laser cavities with nonuniform dielectric along the propagating direction of the cavity, while also taking the open nature of the resonator into account when computing its modes. This strategy allows one to study effects such as dynamic shifts in modal spectrum due to pumping, mode competition, and the addition of external air gap to ﬁne tune the lasing mode wavelength. We also found that while overall blue and red shifts in the spectra can be understood robustly as arising from changes in the intensity weighted refractive index, additional shifts (less than 1 meV) also exist, which are controlled sensitively by the material and geometrical modeling of the resonator.
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