2.1

Active Media Dynamics, Generalized Maxwell-Bloch Equations

Dynamics of interaction between active media and electromagnetic field in the dipole approximation can be described by the Maxwell-Bloch equations, sometimes called the optical Bloch equations.²⁰ Considering generally multimode oscillation, they couple together the electric field intensity E_(i), the electric dipole moment density P_(i) of i-th mode, and the population inversion variables N_q, where index q defines carrier channel or type of the optical transition. In the case of spin-polarized lasers q ∈ {+, −}, referring to different spin projections of conduction band electrons. One of the advantages of optical Bloch equations related to modeling of an anisotropic laser structures such as birefringent spin-V(E)CSELs is respecting of vectorial nature of the electric field intensity and the macroscopic polarization P. As shown in Fig. 1 (left), laser eigenmodes are generally not collinear with the Jones vector characterizing active media eigenpolarization. It is due to optical anisotropies present in a laser cavity and due to anisotropic nature of laser active media. In fact, it means that in a case of such non-collinearity only fraction of incoming electromagnetic field is amplified during stimulated emission.

Mathematically, it is realized by projecting the electric field intensity vector of laser eigenmode on directions of the active media Jones source vectors.²¹ Interaction of multiple modes of electromagnetic field with the spin-polarized carrier densities N_± in an anisotropic laser cavity is described by the generalized Maxwell-Bloch equations¹⁸

where the first of them, Eq. (1), describes the time-dependece of macroscopic polarization P_(i) induced by interaction of the field mode E_(i) with both carrier channels. The macroscopic polarization decays with the rate Γ. Parameter δ represents the spectral detuning and μ is the off-diagonal matrix element of the electric dipole operator. Eq. (2) describes dynamics of the population inversions N_± pumped with the rates _γN_0±, where γ stands for the population decay rate coefficient. Spin-mixing processes can be also described by the effective coefficient γ_s. Equation (3) connects active media to the rest of the laser structure by describing propagation of eigenmode of the frequency ω in entire laser cavity and its dynamics. Decay of laser eigenmodes is described by κ which is defined as 1/τ_ph, where τ_ph stands for the photon lifetime inside cavity. Optical properties of the cavity are described by the permittivity ε, which is generally different for each layer.

As shown in Fig. 1, non-orthogonality of the Jones source vectors is caused by the presence of linear gain anisotropy in semiconductor QW which is induced by reduced crystallographic symmetry at quantum well/barrier interfaces.²² In order to evaluate the exact form of Jones source vectors in an active media with linear gain dichroism, let us define the dipolar transition matrix elements Π_x′, Π_y′ along axes (see Fig. 4). We introduce parameter 1 – Δ which quantifies linear gain anisotropy.¹⁸ It follows that two orthogonal dipolar matrix elements Π_x′, Π_y′ are linked together as Π_y′ = ΔΠ_x′. It means that the dipolar interaction has different strength along x′ and y′. In the reference frame defined by basis axes x = [100] and y = [010], the Jones source vectors characterizing optical transitions in (+) and (−) channels are¹⁸

Note that for Δ = 1, which corresponds, according to our convention, to vanishing linear gain anisotropy, the Jones source vectors describe orthogonal circularly-polarized transitions σ₊ and σ₋, as shown in Fig. 1. For Δ ≠ 1 vectors are not orthogonal to each other, resulting in a strong mode coupling between laser eigenmodes.

Figure 4.

(left) A simplified structure of spin-VECSEL containing 12-QW active region with gain anisotropy used for application of proposed modeling tools. (right) Used reference frame. Source of linear birefringence is localized in two capping layers.

Note that the laser eigenmodes E_(i) are determined not only by properties of active media such as spin polarization P_s or linear gain dichroism Δ, but additionally by optical properties of entire laser cavity.¹⁸ In order to find such laser eigenmodes we will introduce the required matrix formalism in the next parts of this paper.

2.2

Amplification Matrix

We will adapt the description of light interaction with semiconductor QWs based on the generalized Maxwell-Bloch equations presented in Sec. 2.1 for a need of matrix method which will enable us for example to find laser eigenmodes. Using slowly varying envelope approximation and adiabatic elimination of the dipole moment density P_(i) valid for Class-A and Class-B lasers,^{20, 21} we simplify Equation (3) into¹⁸

where the first term represents cavity losses and the second term represents interaction of »-th mode of the electromagnetic field E_(i) with carriers N₊ and N₋. For simplicity, we use notation for the interaction constant.¹⁸ Next, we consider the general electric field intensity vector E_(i) → E. Using methods of tensor calculus, it can be shown that Equation (5) can be re-expressed as^{18, 21}

where we have introduced the gain operator . Note that we are searching for matrix description of monochromatic light passing through an anisotropic QW. For this reason, we go from time to space domain using is speed of considered electromagnetic wave inside QW of the refractive index n. Since a phenomenological description of cavity losses takes into account all of loss mechanisms of photons in a resonator, we are allowed to neglect the loss term during short interval in which photons pass through QW. We can write

which can be solved by conventional methods under the assumption that population inversion is fixed to its threshold value. For a QW of the thickness W we have

where E_out and E_in are intensities of the electric field outgoing from and incoming to QW, respectively. Due to very small thickness of QW we use a linear approximation valid also for arbitrary operator , if , where Î stands for the unity operator. We obtain the operator expression connecting the electric field components outgoing from and incoming to single QW

where it is useful to re-define the gain tensor as :¹⁸

and introduce the gain coefficient g₀¹⁸

where N = N₊ + N₋. We may represent unity operator using the 2 × 2 unity matrix I. Additionally since we are considering semiconductor active media, we should use the Henry’s coefficient α to describe changes in the real part of the refractive index due to pumping of the carriers into QW.²³ Finally, using Eqs. (10) and (11), Eq. (9) becomes¹⁸

where T_μμ are the 2 × 2 components of amplification matrix T and where index μ ∈ {u, d} defines up- and down-travelling amplitudes. In the framework of the scattering matrix formalism, we can describe light amplification in a single semiconductor QW as¹⁸

The amplification matrix T relates amplitudes outgoing from and incoming to QW.

2.3

Propagation in Optical Cavity, Extraction of Laser Eigenmodes

In order to model emission from a V(E)CSEL, it is necessary to describe also propagation in the rest of the laser cavity. We devide the structure into three blocks:^{17, 18} the upper Bragg mirror (or external mirror in the case of 1/2-VCSEL), the active zone and the lower Bragg mirror as shown in Fig. 2. Let us begin with the description of light interaction in the semiconductor multiple QW (MQW) active region. Respecting the notation in Fig. 2, we relate amplitudes interacting with active zone using the amplification matrix T derived in Sec. 2.2 in the following way¹⁸

Figure 2.

The approach used for solving the problem of light propagation and amplification in a complex anisotropic V(E)CSEL or spin-V(E)CSEL. Laser structure is devided into three parts which are described separately.

where also off-diagonal submatrices T_ud and T_du appear due to reflection events in MQW.¹⁸ In the framework of scattering matrix formalism the off-diagonal elements stand for reflectivity matrices. We use scattering matrices S⁽¹⁾ and S⁽²⁾ to describe light propagation in upper and lower mirrors respectively. Definition of S-matrices is presented in Appendix A. According to Fig. 2 we can write¹⁸

where we use the fact that electromagnetic field radiates only out of the structure. In the next step, we solve Eqs. (14), (15) and (16) to obtain emitted amplitudes . Resulting system of four linear equations is written in the matrix form¹⁸

For a nontrivial solution of this system of algebraic equations, which corresponds to laser resonance, the following condition for total matrix must be fulfilled^{17, 18}

Note that, in practice, it is equal to finding the resonant wavelength λ and the threshold value of gain coefficient g₀, for which laser gain exactly compensates cavity losses. Once we find the resonant wavelength and the threshold value of the gain coefficient, Eq. (17) is used to determine the polarization state of the laser eigenmodes and the electric field distribution inside a cavity.

2.4

Standing-Wave Pattern Calculation

Procedure for extraction of laser eigenmodes of complicated anisotropic laser structures presented in Sec. 2.3 turns out to be very useful in determining several steady-state properties of spin-V(E)CSELs such as distribution of electromagnetic field inside laser cavity. Our method for the calculation of standing-wave pattern is described in Fig. 3. The important point is that the field in the entire laser structure can be obtained using emitted amplitudes. The procedure is following: i) we calculate laser eigenmode together with resonant wavelength and polarization state, ii) field is calculated for each z-coordinate using transfer or scattering matrices of each component of laser structure from both sides always up to the last QW (in the sense of considered direction of propagation). Note that in the active zone we obtain two sets of electromagnetic-field components originating in propagating field using transfer matrices from both sides of laser. For this reason iii) we average these two fields. It is necessary to perform the averaging because in the present approach each QW is approximated as an infinitesimally thin dipole layer placed at the center of QW film. Consequently, the field is not amplified smoothly but by steps, which causes certain inaccuracy. It is worth noting that the method is polarization-sensitive. Morover, one can apply recursive S-matrix algorithm¹⁸ to calculate exact field distribution in MQW system.

Figure 3.

The method for calculation of electromagnetic-field distribution inside multi-QW spin-V(E)CSEL. Field components in Bragg mirrors or generally in passive laser components are calculated directly using emitted amplitudes. On the other hand, field inside multi-QW region is obtained as an average of fields given by both emitted amplitudes.

2.5

Rate Equations of Two-Mode Spin-V(E)CSEL With Linear Gain Dichroism

Let us now limit our considerations to dual-mode operation of spin-polarized V(E)CSEL with linear gain dichroism only and neglected birefringence. Such oscillation regime can be described in the limit of Class-A and Class-B lasers by Eqs. (2) and (5), where i ∈ {1, 2}. However, it is impractical for example to solve time-dependence or stability of modes due to mathematical structure of the generalized Maxwell-Bloch equations, e.g. the presence of scalar product. For this reason we derive the rate equation model describing dynamics of anisotropic two-mode spin-polarized V(E)CSEL directly from Eqs. (2) and (5).

We introduce normalized intensities⁸ Ĩ_(1,2) = I_(1,2)/2I_sat, where I_(1,2) are the optical intensities of laser eigenmodes and I_sat is the saturation intensity. We define new population variables⁸ and , where . It can be shown that we may consider N, m ’eigenpopulations’ of 4-level system” in Fig. 1. We find the following system of equations²⁴

where are the effective optical gain coefficients and the cross-coupling coefficients,¹⁸ respectively. Symbol denotes coefficient describing coupling between i-th mode and N_± carrier population inversion, where X ∈ {β, θ}. Note that eigenpolarizations can be obtained in the absence of any additional anisotropies in laser cavity such as the linear birefringence simply by finding eigenvectors of a gain tensor in Eq. (10).¹⁸ Overall pumping rate is quantified using _γN₀, where N₀ stands for the unsaturated population inversion.²¹ Spin polarization of pump is controlled by the parameter P_∈ and so-called spin flip rate coefficient is defined as γ + 2γ_s.⁸

Eqs. (19-22) describe coupling of two modes E_(1,2), generally differing in wavelength and polarization, to carrier population inversions N_±. Such coupling between modes increases with increasing linear gain dichroism quantified using the parameter Δ.¹⁸ Note that in the absence of gain dichroism (and birefringence), which is equivalent to consider Δ = 1, there is no cross-coupling. It means that and consequently (21), (22) become equivalent to (27), (28) (see Appendix B). One can see that also the rest of rate equations of our model become formally equivalent to spin flip model in the absence of linear gain anisotropy. In this sense, Eqs. (19-22) form the generalization to spin flip model where amplitude anisotropies are described by effective parameter γ_a and are not localized.^{8, 10} It should be noted that our approach to solve time-dependence of spin-V(E)CSELs can be adapted to include also linear birefringence that can be localized in arbitrary layer of laser structure. In the case of birefringent spin-V(E)CSEL structure, it is not possible to find eigenmodes as a gain tensor eigenvectors anymore and numerical methods must be used instead. It means that the calculation of effective gain coefficients and cross-saturation coefficients is much more complicated. However, this is out of the scope of this article.

3.1

Electromagnetic-Field Distribution Calculation

Let us now apply method of electromagnetic-field distribution calculation described in Sec. 2.4 to above-mentioned 12-QW VCSEL structure. Fig. 5 shows the resulting standing-wave pattern inside 1/2-VCSEL for Δ = 0.95, δn = 0.025 and the effective degree of spin polarization P_s = 0. The polarization of the optical eigenmode is linear. Since axes of linear gain anisotropy and linear birefringence are not aligned, a vector of electric field intensity is oriented in quite general direction. We would obtain E_x = E_y in each point of spin-VECSEL, if we took δn = 0. On the other hand, E_x component increases with increasing linear birefringence parameter δn. We note that while searching for anisotropic spin-VECSEL eigenmodes, we usually find multiple sets of two transversal modes. Those sets are equidistant. We do not consider simultaneous oscillation of these two transversal modes in this subsection. Consequently, what we present is the field distribution of a single transversal mode.

Figure 5.

Electromagnetic-field distribution inside 1/2-VCSEL with linear gain dichroism and linear birefringence. Orthogonal directions are not equivalent due to optical anisotropies inside the laser cavity. We see that the wavelength of optical eigenmode lays inside photonic bandgap of DBR and thus electromagnetic field exponentially decays inside the Bragg mirror.

In the next case (see Fig. 6), we consider also non-zero effective spin polarization degree P_s. We see that using quite large value of P_s, we are able to almost compensate the effects of linear optical anisotropies.^{11, 16} Polarization of emitted field is very close to perfectly circular one, which can be deduced from the values of the Stokes vector describing polarization state of emitted light.

Figure 6.

Electromagnetic-field distribution inside 1/2-VCSEL with linear gain dichroism and linear birefringence and pumped using spin-polarized current. Large effective spin polarization of conduction band electrons inside QWs is able to almost compensate the effects of linear optical anisotropies.

3.2

Dynamical Properties of Spin-V(E)CSELs

This part of our contribution is dedicated to analysis of time-dependent phenomena in spin-V(E)CSELs. We study transient behavior of both laser eigenmodes under different conditions and polarization dynamics using time-dependent Stokes vectors components of emitted light. All the calculations are performed for spin-polarized 1/2-VCSEL with external cavity described in the previous subsection and shown in Fig. 4 (left). We emphasize here the effects of anisotropic properties of the laser cavity and active media such as the linear gain anisotropy and the spin polarization of conduction band electrons in QWs. Note that considering dynamical properties of spin-lasers linear birefringence is out of the scope of the present work.

The analysis is based on Eqs. (19-22) which are solve using conventional numerical Runge-Kutta methods.²⁵ We use following values of parameters for simulations:⁶ the spin lifetime , the carrier lifetime τ = 1/γ = 1 ns and the photon lifetime τ_ph = 1/κ = 10 ns. Note that we are able to calculate the electric field decay rate coefficient κ or equivalently the photon lifetime τ_ph for arbitrary optical cavity using the scattering matrix formalism. It can be derived using Fourier analysis that the photon lifetime in a laser cavity is given by the cavity’s transmission line shape.²⁶ Normalized pumping parameter is fixed at value N₀ = 5, it means five times the threshold pumping rate. There are two controlled parameters of simulations, the pump polarization P_∈ and the linear gain dichroism parameter Δ.

In our first result (see Fig. 7), we consider realistic experimental situation of spin-polarized VECSEL with the gain dichroism characterized with Δ = 0.9. We solve time-dependence of both laser eigenmodes for several values of pump polarization P_∈. In the first case, for P_∈ = 0, eigenmode E₍₂₎ quickly decays to zero. A possible physical interpretation is following: since there is zero spin polarization of electrons in QWs, two transversal linearly-polarized modes along [110] and axes can oscillate in a cavity.¹⁸ This can be deduced from gain tensor defined in Eq. (10). These two possible modes interact with joint carrier reservoir described by carriers density N = N₊ + N₋. However, they have different gain coefficient due to linear gain dichroism, which results in instability of mode with larger threshold population inversion.¹⁰ Such degeneracy of carrier reservoirs is lifted by non-zero spin pumping (P_∈ ≠ 0). As shown in Fig. 7, oscillation of both modes is possible for moderate values (P_∈ = 0.2, P_∈ = 0.4, P_∈ = 0.6) of pump polarization. In this sense, spin pumping is able to compensates the effects of linear gain anisotropy. On the other hand, if pump polarization is too large, only single mode oscillation regime occurs. The reason is that when we reduce the threshold pumping rate of preferential mode using spin pumping, we increase needed pumping rate for weaker mode. This effect is probably even stronger in presence of linear gain dichroism.⁶

Figure 7.

Time-dependence of two laser eigenmodes of spin-VECSEL calculated with fixed value of linear gain dichroism parameter Δ = 0.9. Pump polarization P_∈ is variable parameter.

Simulations presented in Fig. 8 consider experimentally rather impractical situation, because in this case linear gain dichroism is variable. It shows that fixed value of pump polarization of about P_∈ = 0.5 is able to compensate (in the sense of two-mode oscillation regime) values of linear gain dichroism coefficient even slightly smaller that Δ = 0.9. We see that preferential mode E₍₁₎ has typical Class-A laser characteristics which is given by values of the carrier and photon lifetimes.^{20, 21} This is not the case for second mode which always oscillates for a short time and then quickly decays to zero. It is caused by negative mode coupling – stronger mode makes the weaker one unstable.

Figure 8.

Time-dependence of two laser eigenmodes of spin-VECSEL calculated with fixed value of pump polarization P_∈ = 0.5. Non-zero pump polarization together with linear gain dichroism result in preference of one mode E₍₁₎ over the other one E₍₂₎.

In the following part we investigate polarization dynamics of spin-VECSELs. For this reason, we define the Jones vector of emitted field as

where J₍₁₎ and J₍₂₎ are the Jones vectors of laser eigenmodes, which can be extracted directly from gain tensor in the absence of additional anisotropies in a laser cavity. Definition of J fulfills the normalization condition J^∗ · J = 1 in a case of orthogonal eigenpolarizations. Using Eq. (23), we are able to calculate components of the Stokes vector describing polarization of emitted laser field in terms of optical intensities.

In order to determine polarization state of emitted field, we solve Eqs. (19-22) to obtain steady-state normalized intensities of laser eigenmodes. Next, we extract J₍₁₎ and J₍₂₎ using gain tensor. In Fig. 9, we show the calculated Stokes vector components S₂ = I_x′ − I_y′, S₃ = I₊ − I₋ defined as intensity difference of linear modes along x′, y′ axes and difference of orthogonal circular polarizations, respectively. Note that axes x′, y′ are principal axes of linear gain anisotropy. Consequently, the Stokes vector element S₂ should be very sensitive to changes in Δ. We see that this claim is confirmed by calculation. It means that S₂ changes more rapidly with changing Δ than with P_∈. On the other hand, the element S₃ should be more sensitive to changes in spin pumping since circular polarizations are eigenpolarizations of spin-laser in the absence of any anisotropies. Again, it is verified by presented model. Note that obtained result depends on dynamical properties of spin-VECSELs such as mode coupling.

Figure 9.

Stokes vector components S₂, S₃ expressed as functions of pump polarization P_∈ and linear gain dichroism paramater Δ. The result slightly differs from the calculations for each of eigenmodes separately.

In Fig. 10, we present results concerning polarization dynamics. We express Stokes vector components as a function of time in interval until laser reaches its steady-state.

Figure 10.

Time-dependence of Stokes vector components calculated for the fixed value of linear gain dichroism coefficient Δ = 0.9 and for several values of pump polarization P_∈.